covering arrays with row limit - university of toronto t-space · 2012. 12. 11. · abstract...

Covering arrays with row limit

by

Nevena Francetic

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

Copyright c© 2012 by Nevena Francetic

Abstract

Covering arrays with row limit

Nevena Francetic

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2012

Covering arrays with row limit, CARLs, are a new family of combinatorial objects

which we introduce as a generalization of group divisible designs and covering arrays. In

the same manner as their predecessors, CARLs have a natural application as combinatorial

models for interaction test suites. A CARL(N ; t, k, v : w), is an N × k array with some

empty cells. A component, which is represented by a column, takes values from a v-

set called the alphabet. In each row, there are exactly w non-empty cells, that is the

corresponding components have an assigned value from the alphabet. The parameter w is

called the row limit. Moreover, any N × t subarray contains every of vt distinct t-tuples

of alphabet symbols at least once.

This thesis is concerned with the bounds on the size and with the construction of

CARLs when the row limit w(k) is a positive integer valued function of the number

of columns, k. Here we give a lower bound, and probabilistic and algorithmic upper

bounds for any CARL. Further, we find improvements on the upper bounds when

w(k) lnw(k) = o(k) and when w(k) is a constant function. We also determine the

asymptotic size of CARLs when w(k) = Θ(k) and when w(k) is constant.

Next, we study constructions of CARLs. We provide two combinatorial constructions

of CARLs, which we apply to construct families of CARLs with w(k) = ck, where c < 1.

Also, we construct optimal CARLs when t = 2 and w = 4, and prove that there exists a

constant δ, such that for any v and k ≥ 4, an optimal CARL(2, k, v : 4) differs from the

ii

lower bound by at most δ rows, with some possible exceptions.

Finally, we define a packing array with row limit, PARL(N ; t, k, v : w), in the same

way as a CARL(N ; t, k, v : w) with the difference that any t-tuple is contained at most

once in any N × t subarray. We find that when w(k) is a constant function, the results on

the asymptotic size of CARLs imply the results on the asymptotic size of PARLs. Also,

when t = 2, we consider a transformation of optimal CARLs with row limit w = 3 to

optimal PARLs with w = 3.

iii

To my Ujka Stanko, the initiator and the most fervent supporter of my

academic work.

iv

Acknowledgements

First of all, I would like to express my sincere thanks to Prof. Eric Mendelsohn and

Prof. Peter Danziger for their unfaltering faith and unbound confidence in me. They

supervised and supported my work, and dedicated many hours to listening and helping

me with numerous problems. Their advice was invaluable. Sincere thanks to all members

of my committee, especially to Prof. David Pike for being my external examiner.

Next, I would like to give thanks to my parents, Miroslav and Branka, sisters, Tanja

and Dunja, and brother-in-law, Kostya, without whose moral support and constant

encouragement this project would have not been possible. I want to thank many new

friends I made in Toronto who made the graduate school and life in this great city lots of

fun. Thanks to Anna Shamaeva for providing many funny stories on daily basis and to

Milka Matejic for being my devoted friend in many adventures.

Sincere thanks to Prof. Mendelsohn, Prof. Danziger, Tanja, Dunja and Anna for proof

reading parts of my thesis.

I would like to express sincere gratitude to NSERC for the financial support, as well

as to the Department of Mathematics and the School of Graduate Studies, University

of Toronto. Special thanks to Ida Bulat for always having good advice and providing

outstanding help.

I would also like to mention and thank the people who inspired a mathematician in me.

Thanks to Prof. Jovan Knezevic for motivating me to study and compete in mathematics.

Sincere thanks to Prof. Mateja Sajna for discovering a researcher in me.

v

Contents

Glossary 1

1 Introduction 4

1.1 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Preliminaries 12

2.1 Miscellaneous definitions related to bounds . . . . . . . . . . . . . . . . . 12

2.2 Block Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Incomplete, Holey, and Double Group Divisible Designs . . . . . . 16

2.3 Coverings and Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Covering Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Probabilistic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Definitions and Examples 24

3.1 Covering arrays with row limit . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Packing arrays with row limit . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Bounds 35

4.1 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 The Schonheim lower bound . . . . . . . . . . . . . . . . . . . . . 37

4.2 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

vi

4.2.2 Uniform Distribution Bound . . . . . . . . . . . . . . . . . . . . . 40

4.2.3 Binomial Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Second Moment Bound for CARLs when w(k) lnw(k) = o(k) . . 46

Asymptotic size of CARLs when row limit is a constant . . . . . 49

4.3 Bounds comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 General Constructions 55

5.1 Algorithmic construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Product construction of CARLs with t = 2 . . . . . . . . . . . . . . . . . 58

5.3 Wilson’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3.1 Applications of Wilson’s Construction . . . . . . . . . . . . . . . 63

5.3.2 Analysis of the constructed objects . . . . . . . . . . . . . . . . . 66

6 Group divisible covering designs with block size four 70

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2 Constructions and notation . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3 g = 2 or 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.4 Constructions of 4−GDCD with a small number of groups . . . . . . . 83

6.4.1 Constructions using Double Group Divisible Designs . . . . . . . 85

6.5 4−GDCD of type gu where g 6= 1, 2, or 6 . . . . . . . . . . . . . . . . . 105

6.6 Another construction method when g ≡ 2 (mod 3) and u ≡ 5 (mod 6) . 110

6.7 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.8 Summary of constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7 Packing arrays with row limit with constant block size 120

7.1 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.1.1 Schonheim upper bound . . . . . . . . . . . . . . . . . . . . . . . 121

7.1.2 Asymptotic size of PARLs with constant row limit . . . . . . . . 123

7.1.3 The Johnson bounds for PARLs and CARLs . . . . . . . . . . . 124

vii

Johnson lower bound for CARLs . . . . . . . . . . . . . . . . . . 127

7.2 Construction of optimal 3−GDPDs from optimal 3−GDCDs . . . . . 128

7.2.1 Examples of optimal 3 − GDCDs which transform into optimal

3−GDPDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.2.2 No edges in excess (leave) graphs . . . . . . . . . . . . . . . . . . 135

7.2.3 Two edges in excess graphs . . . . . . . . . . . . . . . . . . . . . 136

7.2.4 One regular excess (leave) graphs . . . . . . . . . . . . . . . . . . 136

7.2.5 Almost one regular excess (leave) graphs . . . . . . . . . . . . . . 138

7.2.6 Optimal 3−GDPDs . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.2.7 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.2.8 Summary of 3-GDCD constructions . . . . . . . . . . . . . . . . . 142

8 Conclusion 145

8.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.2 Future work and open questions . . . . . . . . . . . . . . . . . . . . . . . 148

A List of missing ingredients 152

B Some optimal 3−GDPDs 155

Bibliography 193

Index 197

viii

Glossary

Notation Description Page

List

(K,λ)−GDD of type gu11 gu22 . . . guss group divisible design 14

BIBD(v, k, λ) balanced incomplete block design 13

C(v, k, t) cover number of a t− (v, k, λ) covering 18

CANλ(t, k, v) cover number of a CAλ(t, k, v) 21

CARL(t, k, v : w) covering array with row limit where λ = 1

when the size N is not essential in the context

25

CARLNλ(t, k, v : w) minimum size of a CARLλ(t, k, v : w) 26

CARLλ(N ; t, k, {v1, v2, . . . , vk} : w) covering array with row limit 24

CARLλ(N ; t, k, v : w) covering array with row limit for which all

components have alphabet size v

25

CAλ(N ; t, k, v) covering array 20

D(k, gu11 gu22 . . . guss ) maximum number of blocks in a k −GDPD

of type gu11 gu22 . . . guss

32

GDCD group divisible covering design 27

GDPD group divisible packing design 32

K −DGDD of type (n, hu11 hu22 . . . huss ) double group divisible design 17

K −HGDD of type (n, hu) holey group divisible design 16

K − IGDD of type (g, h)u incomplete group divisible design 16

1

Glossary 2


List

K set of block sizes of a design or a GDCD 27

N size of a CARL, i.e. number of rows in the

array

25

OAλ(t, k, v) orthogonal array 21

PARL(N ; t, k, v : w) packing array with row limit for which all com-

ponents have alphabet size v

31

PARLNλ(t, k, v : w) maximum size of a PARLλ(t, k, v : w) 31

PARLλ(N ; t, k, {v1, v2, . . . , vk} : w) packing array with row limit 31

PBD(v,K, λ) pairwise balanced design 13

TDλ(k, n) transversal design 15

[m,n] {m,m+ 1, . . . , n} 12

λ index of a CARL or a design 25

dxe ceiling function 12

bxc floor function 12

g group size of a GDD or a GDCD 27

k degree, i.e. number of columns, of a CARL or

block size of a design

13, 25

t− (v,K, λ) design a t-design 13

t− (v, k, λ) covering 18

t strength of a CARL or a design 25

u usually, number of groups of a GDD or a

GDCD

27

v alphabet size of a component of a CARL or

the number of elements in a design

13, 25

w∗ mandatory block size of a PBD 13

Glossary 3


List

w row limit of a CARL or a PARL 25, 31

y = x(1± ε) means that y ∈ (x− εx, x+ εx) 12

|V | order of a set V 12

C(k, gu11 gu22 . . . guss ) minimum number of blocks in a k − GDCD

of type gu11 gu22 . . . guss

28

excess graph multi-graph whose edges are incident to the

pairs of elements contained in more than one

block of a GDCD, counting multiplicities

29

leave graph graph whose edges are incident to the pairs of

elements not contained in any block or a group

of a GDCD

32

Chapter 1

Introduction

Covering arrays with row limit, CARLs for short, are a new family of combinatorial

objects which we introduce as a generalization of group divisible designs and covering

arrays, two well-known families of objects in combinatorial design theory. Group divisible

designs, GDDs for short, were introduced as statistical experiment designs [17,51]. On

the other hand, covering arrays model test suites in software testing [11]. The motivation

for the definition of CARLs stems from a propensity to adapt the definition of a covering

array to model an interaction test suite in, for example, pharmacology where one has to

limit the number of drugs administered to an individual at once. Our study of CARLs

shows that they sometimes behave as GDDs and under different conditions they behave

as covering arrays. As well as being related to GDDs and covering arrays, CARLs can

be interpreted as an edge covering problem in the graph theory.

are contributing new results to the graph edge covering problem.

As with any covering problem, two central questions in the study of CARLs are: what

the minimum size of a covering is and how to construct a covering of the minimum size.

We start by developing upper and lower bounds on the size of a CARL. We then consider

constructions of optimal covering arrays with row limit four. Finally, we study a related

family of objects called packing arrays with row limit, PARLs for short.

4

Chapter 1. Introduction 5

0 0 − 1 − 00 − 1 1 1 −0 1 0 − − 10 − 0 0 0 −1 0 1 0 − −1 − 0 − 1 01 − 1 − 0 11 1 − 1 − 1− 0 0 1 0 −− 1 1 − 1 0− 0 − 0 1 1− 1 − 0 0 0

Figure 1.1: An optimal CARL(12; 2, 6, 2: 4).

We state the formal definitions of GDDs and covering arrays in Chapter 2 and of

CARLs and PARLs in Chapter 3. Here we introduce CARLs by an example. Then we

discuss the applications and the characteristics of group divisible designs and covering

arrays which lead to the definition and study of CARLs.

Example 1.1. A CARL is a two dimensional array. For example, a CARLλ=1(N =

12; t = 2, k = 6, v = 2: w = 4) is given in Figure 1.1. The parameter k = 6 denotes the

number of columns. In a testing, each column represents a component to be tested which

may have a certain number of levels. In our example, all components/columns have v = 2

levels, which is represented by the alphabet set {0, 1}. Some cells in the array contain

a ‘−’, which means that the cell is empty; no alphabet value is assigned to it. The row

limit, w, denotes the number of non-empty cells per row. In our example, there are w = 4

non-empty cells in each row. The parameter t denotes the strength and the parameter λ

is index, and together they define the covering property: any t columns contain every

possible combination of alphabet symbols at least λ times. In our example, any two

columns contain (cover) at least once each of the pairs (0, 0), (0, 1), (1, 0), and (1, 1).

When the index λ = 1, it is usually omitted in the notation. Finally, the parameter N

denotes the size, that is, the number of rows in the array. Usually, values of t, k, v and


w are given, and the objective is to minimize the number of rows in order to satisfy the

covering property. For the values of the parameters given in our example, the smallest

possible number of rows is N = 12 (which follows from Theorem 4.2), and hence the array

in our example is optimal.

A covering array is a CARL without empty cells, that is the row limit, w, equals

the number of columns, k. The relationship between CARLs and GDDs is not obvious.

When strength t = 2, a group divisible design is equivalent to a CARL in which every pair

of columns contains every possible combination of alphabet values exactly λ times. The

other difference is that a GDD is usually defined as a triple of sets, where as a CARL is

defined as an array. We further discuss this relationship in Chapter 3.

The study of group divisible designs, GDDs was started in 1942 when Bose suggested

a notation for them [4, 14]. The name, group divisible designs, was established about

ten years later in 1952, when two articles studying group divisible designs as statistical

experiment designs were published [17,51]. In [51], a GDD was defined as an experimental

design in which there are gu treatments, separated into u groups of g treatments. The

objective was to design a test of k treatments at a time (i.e. to find blocks of size k), such

that any two treatments in the same group are tested λ1 times together, and any two

treatments in two different groups appear in λ2 tests together. Today, in the standard

definition of GDDs, λ1 = 0, and we denote them by (k, λ2) − GDD of type gu (see

Definition 2.1). Since the 1950’s, more than five hundred papers on GDDs have been

published. In addition, there are many generalizations of GDDs. In Chapters 6 and 7, we

use incomplete group divisible designs, IGDDs, holey group divisible designs, HGDDs,

and double group divisible designs, DGDDs, in our constructions of CARLs with constant

row limit.

It is not difficult to determine the number of blocks in a GDD when it exists. Indeed,

following the above definition of a GDD, since any pair of elements in two distinct groups

appears in exactly λ2 blocks of size k, one can determine the necessary conditions on g, u,


and k for the existence of a GDD. However, the larger the block size is, the harder it

is to prove that the necessary conditions are also sufficient. Currently, there are a few

known families of GDDs with block size k ≥ 6 [14].

Covering arrays were first applied in the zero-error noisy channel communication

problem [33] and in compressing inconsistent data [32]. With the rapid development of

the software industry, covering arrays have become well known as interaction test suites

in ‘combinatorial’ or ‘all-pair’ black-box testing. Covering arrays are used for detection

of faulty interactions between components. Empirical studies show that most of the faulty

interactions are detected by testing pair-wise interactions [34] which spurred the research

into covering arrays with strength t = 2. With increased complexity of devices, shorter

development time, and higher expectations in quality, there has been an increase in the

demand for optimization of the interactions testing process and hence for better covering

arrays. The website www.pairwise.org currently lists thirty six different tools for finding

a covering array including a number of commercial products.

Unlike GDDs which may exist only when the necessary conditions are satisfied, the

existence problem of covering arrays is trivial; they always exist since for each t-tuple of

elements, we can dedicate a row to cover it. However, determining the minimum number

of rows is a difficult problem [11]. There are many bounds on the size of a covering array,

but for a given a set of parameters, the optimal size of a covering array is not known in

general [11,14].

A natural generalization of GDDs and covering arrays is to keep the covering property

of covering arrays and the variable block size of GDDs. The block size of GDDs

corresponds to a new parameter for covering arrays, the row limit, which represents the

number of non-empty cells in a row (see Definition 3.1).

There are number of possible combinatorial testing applications of CARLs. In medical

or pharmaceutical testing, it might be inadvisable or unethical to administer more than

a certain number of medications at once to the subjects in a study. In a chemical or

www.pairwise.org


new-material testing, it might be suitable to limit the number of compounds tested at

once if there is a concern that combining them together may trigger difficult to control

or undesired interactions. Another application may be that there is a constraint on the

amount of available resources per test, such as available bandwidth. Furthermore, it

might be easier to find a set of subjects for a study which satisfies a specific smaller set of

requirements. Also, setting up the next test case might be costly or time consuming, and

changing a smaller set of parameters might be more efficient.

Mathematically, analogous to covering arrays, CARLs always exist. Two central

questions are determining the optimal size of a CARL and constructing a CARL of the

optimal size, or a size close to optimal. In our study, we take the values of the strength

t, and the size of the alphabet to be given constants, but we consider the row limit

w = w(k) as a positive integer valued function of k, where k is the number of columns.

The two extremal cases are when w(k) = k, which corresponds to covering arrays, and

when w(k) is a constant function, which corresponds to a covering version of GDDs

with strength t. We already know that in the first case, the size of a covering array is

difficult to determine [11]. In the second case, we give an asymptotically optimal bound

(cf. Theorem 4.7). This opens the question of size and behaviour of CARLs for a general

function w(k).

Given that CARLs generalize both GDDs and covering arrays, they also inherit the

characteristics and constructions of both, which is reflected in an ample number of known

results listed in Chapter 2 (Preliminaries), which we use in our study afterwards. These

two areas of study also suggest two different and somewhat conflicting notations for

CARLs. We give two equivalent definitions of this new family of combinatorial objects. In

the array representation, these objects are called CARLs and their notation is consistent

with the notation for covering arrays. On the other hand, in the set version, these objects

are called group divisible covering designs, GDCDs, and they follow the notation of GDDs.

Usually one thinks of a GDD as a design with a fixed block size and variable numbers of


groups and group sizes. Therefore when the row limit is a constant and strength t = 2,

we use the GDD notation; otherwise, we use the CARL notation. Unfortunately, the

notations for GDDs and covering arrays are not consistent (see Definitions 2.1 and 2.3).

In fact, they sometimes use the same symbols for different parameters. In particular, k

is the number of columns of a covering array while the same k also represents the block

size of a GDD; v is the alphabet size of a column of a covering array but v is the total

number of elements in a design in general. To mitigate the confusion, we refer a reader to

the glossary of terms and notations.

Another possible generalization of GDDs and orthogonal arrays is to alter the condi-

tions so that every t-tuple of elements is covered by at most λ blocks or equivalently rows.

The array version of this problem is called a packing array with row limit, PARL (see

Definition 3.5), and the set version when t = 2 is called a group divisible packing design,

GDPD. We primarily study CARLs; however, some of our results and techniques for

construction of CARLs directly imply or are adaptable to PARLs. These results are

discussed in Chapter 7.

1.1 Outline of Thesis

In Chapter 2 (Preliminaries) we cite the definitions and some known results for a number

of block designs used later in our study. In Chapter 3 (Definitions and Examples) we

define the main objects of the study, CARLs and GDCDs, as well as PARLs and their

set version GDPDs. We also give some examples of these objects and the equivalence

relationships between their different forms. In addition, we discuss the graph version of

these problems and define the excess and leave graphs.

In Chapter 4 (Bounds) we study upper and lower bounds on CARLs. We start with

deriving an analog of the Schonheim lower bound for CARLs. We then obtain several

upper bounds on the size of a CARL using two probabilistic methods. We then strengthen


the results of one of the probabilistic methods for two other classes of functions of the

row limit w(k).

In Chapter 5 (General Constructions) we give an algorithm for construction of a

CARL. We also consider two combinatorial constructions of CARLs, one of which is for

any strength t, and the other is only for CARLs of strength t = 2. We also apply these

constructions to obtain several families of CARLs whose size is below the upper bounds

given in Chapter 4. Results of Chapters 4 and 5 will appear in [18].

In Chapter 6 (Group divisible covering designs with block size four), we develop

constructions of optimal CARLs with strength t = 2 and row limit w = 4. Since the

strength is equal to two and the row limit is a constant, we construct group divisible

covering designs with block size four rather than CARLs. All constructions in this chapter

are based on two types of ingredients: essential and auxiliary. The essential ingredients

are used multiple times in a construction, and they have to meet the lower bounds in

order for the construction to be, in the worst case, close to optimal. On the other hand,

the auxiliary ingredients are used only once in the construction, and the result of the

construction differs from the lower bound by the same number of blocks as the number of

blocks the auxiliary ingredient differs from the lower bound. We give an optimal solution

for all essential ingredients with less than fifty nine starter blocks, and we construct many

families of optimal 4−GDCDs. Moreover we prove that with some possible exceptions,

there exists a constant δ such that for any set of parameters, an optimal 4 − GDCD

differs from the lower bound by at most δ blocks. This study is going to appear in [19].

The results regarding bounds on the size of CARLs with constant row limit in

Chapter 4 and the constructions of such CARLs in Chapter 6 can be adapted to the

packing version of this problem. In Chapter 7 (Packing arrays with row limit with constant

block size), we derive these results for PARLs. Next, we strengthen the Schonheim upper

bound for PARLs by a generalization of the first Johnson’s bound which holds for some

families of objects. Finally, we apply similar constructions as in Chapter 6 to construct


optimal GDCDs with row limit three which transform into optimal GDPDs, and vice

versa. The study of transformable GDCDs is not completed at this time. However, we

do construct all optimal GDPDs with block size three with only eight possible exceptions

(see Theorem 3.1), some of which are not necessarily transformable into optimal GDCDs.

After finalizing these results, they are going to appear in [20].

Chapter 8 (Conclusion) contains a summary of the thesis results and a discussion

of some open questions and possible further directions of study related to CARLs and

PARLs.

Chapter 2

Preliminaries

2.1 Miscellaneous definitions related to bounds

For completeness, we state some standard definitions here: bxc is the floor function

whose value is the greatest integer smaller than or equal to x ∈ R; dxe is the ceiling

function whose value is the smallest integer greater than or equal to x ∈ R. We say that

y = x(1 ± ε) if y ∈ (x − εx, x + εx); and [m,n] = {m,m + 1, . . . , n}, where m,n ∈ Z,

m ≤ n. If V is a set, |V | denotes the order of V , that is the number of elements in V .

The o and O notations are used extensively in stating asymptotic results but the following

definitions are less well known. Let f, g : R→ R. Then,

• f(x) = ω(g(x)) if for every M > 0, there exists x0 such that for all x ≥ x0,

M g(x) ≤ f(x);

• f(x) = Θ(g(x)) if there exist M1, M2 and x0 such that for all x ≥ x0, M1 g(x) ≤

f(x) ≤M2 g(x).

2.2 Block Designs

The constructions presented in Chapters 6 and 7 depend on many well known designs.

Here we give the definitions and existence results which we will use later. We follow the

12

Chapter 2. Preliminaries 13

notation given in [14].

A t− (v,K, λ) design [14, 50] is a pair (V,B) where V is a v-set of elements, and B is

a collection of subsets of V , called blocks, such that

1. for every B ∈ B, |B| ∈ K, and

2. every t-subset of V is contained in exactly λ blocks.

It is called a t− design for short. A subset P ⊂ B is called a parallel class if the blocks in

P form a partition of V . If there exists a partition of B into parallel classes, then (V,B)

is called a resolvable t-design.

We now consider some specific types of t-designs. A pairwise balanced design

PBD(v,K, λ) is a 2− (v,K, λ) design. A balanced incomplete block design BIBD(v, k, λ)

is a PBD(v,K, λ) such that K = {k}.

Theorem 2.1. [26] There exists a BIBD(v, 4, 1) if and only if v ≡ 1, 4 (mod 12), v ≥ 4.

Let v and λ be positive integers, let K be a set of positive integers and let w ∈ K.

A PBD(v,K, λ) has a mandatory block size w if it must contain a block of size w. To

indicate that there is exactly one block of size w, we use w∗ in the list of block sizes.

Theorem 2.2. [45] There exists a PBD(v, {4, w∗}, 1) if and only if v ≥ 3w + 1 and

either v ≡ 1, 4 (mod 12) and w ≡ 1, 4 (mod 12), or v ≡ 7, 10 (mod 12) and w ≡ 7, 10

(mod 12).

In Chapter 1, we mentioned that a group divisible design (GDD) is equivalent to a

CARL for which in any two columns, every combination of alphabet symbols is covered

exactly λ times, where λ is the index. Here we give a formal definition of a GDD.

Definition 2.1. A group divisible design, for short a GDD, with index λ is a triple

(V,G,B), where V is a set of elements, G is a partition of V into subsets, called groups,

and B is a collection of subsets of V , called blocks, such that


1. for every G ∈ G and every B ∈ B, |G ∩B| ≤ 1, and

2. every pair of elements of V belonging to two distinct groups is contained in exactly

λ blocks.

A (K,λ)−GDD of type gu11 gu22 . . . guss is a group divisible design (V,G,B) such that

|V | =∑s

i=1 giui, there are ui groups of size gi, for i ∈ [1, s], and K is the set of block

sizes, that is for every B ∈ B, |B| ∈ K.

There are several notation rules regarding GDDs which also hold for designs related

to GDDs. A (k, λ)−GDD is a GDD for which K = {k}. If λ = 1, λ and the parenthesis

are usually omitted in the notation, thus a k −GDD is a GDD for which K = {k} and

λ = 1.

We now state some known results about group divisible designs, which we rely on

later.

Theorem 2.3. [55] There exists a (3, λ)−GDD of type gu if and only if u ≥ 3, λg(u−1) ≡

0 (mod 2), and λg2u(u− 1) ≡ 0 (mod 6).

Theorem 2.4. [5] A (4, λ)−GDD of type gu exists if and only if u ≥ 4, λg(u− 1) ≡ 0

(mod 3), λg2u(u− 1) ≡ 0 (mod 12), except when (g, u, λ) ∈ {(2, 4, 1), (6, 4, 1)}.

Theorem 2.5. [15] There exists a 3−GDD of type gum1 for non-negative integers g, u

and m if and only if the following conditions are satisfied:

1. if g > 0, then u ≥ 3, or m = g and u = 2, or u = 1 and m = 0 or u = 0;

2. m ≤ g(u− 1) or gu = 0;

3. g(u− 1) +m ≡ 0 (mod 2) or gu = 0;

4. gu ≡ 0 (mod 2) or m = 0;

5. 12g2u(u− 1) + gum ≡ 0 (mod 3).


Theorem 2.6. [22] There exists a 4−GDD of type 2um1 for all u ≥ 6, u ≡ 0 (mod 3)

and m ≡ 2 (mod 3), 2 ≤ m ≤ u− 1 except when (u,m) = (6, 5), and possibly except for

(u,m) ∈ {(21, 17), (33, 23), (33, 29), (39, 35), (57, 44)}.

Theorem 2.7. [22] There exists a 4−GDD of type 4um1 for all u ≥ 6, u ≡ 0 (mod 3)

and m ≡ 1 (mod 3) with 1 ≤ m ≤ 2(u− 1).

Theorem 2.8. [23] There exists 4−GDD of type 6um1 if u ≥ 4, m ≡ 0 (mod 3), and 0 ≤

m ≤ 3(u− 1), except possibly when (u,m) ∈ {(7, 15), (11, 21), (11, 24), (11, 27), (13, 27),

(13, 33), (17, 39), (17, 42), (19, 45), (19, 48), (19, 51), (23, 60), (23, 63)}.

Theorem 2.9. [48] Let g = 8 or 16. There exists a 4−GDD of type gum1 when either

(u,m) = (3, g) or u ≥ 6, u ≡ 0 (mod 3), and m ≡ g (mod 3) with 0 < m ≤ g2(u− 1).

Theorem 2.10. [22] There exists a 4 − GDD of type 12um1 for all u ≥ 4 and m ≡ 0

(mod 3) with 0 ≤ m ≤ 6(u− 1).

Finally, we define a special kind of GDDs called transversal designs. We also introduce

orthogonal arrays.

Definition 2.2. A transversal design TDλ(k, n) is a (k, λ)−GDD of type nk.

When λ = 1, it is usually omitted from the notation.

Theorem 2.11. [50, Theorem 6.51] There exists a TD(4, n) if and only if n 6= 2, 6.

An orthogonal array OAλ(t, k, v) is a λvt × k array, A, with entries from a v-set V

such that any λvt × t subarray of A contains every possible ordered t-tuple of elements

from V exactly λ times. Again, if λ = 1, it is omitted in the notation. There is a well

known equivalence relation between a set of k mutually orthogonal latin squares of order

n, a transversal design TD(k + 2, n) and an orthogonal array OA(2, k + 2, n) [50]. The

same equivalence relation holds between GDCDs and CARLs.

In Section 5.3, we rely on the following result regarding orthogonal arrays which we

use in constructions of CARLs.


Theorem 2.12. [7] Let q be a prime power and t < q be an integer. There exists an

orthogonal array OA(t, q+1, q). Moreover, if q = 2n, n ≥ 2, there exists an OA(3, q+2, q).

2.2.1 Incomplete, Holey, and Double Group Divisible Designs

Incomplete, holey and double group divisible designs are generalizations of group divisible

designs which we use extensively in constructions of GDCDs. Here we give their definitions

and some existence results.

An incomplete group divisible design IGDD is a quadruple (V,H,G,B) where V is a

set of points and H is a subset of V , called a hole. Moreover, G is a partition of V into

disjoint groups, and B is a collection of subsets of V , called blocks, such that

1. for every B ∈ B and G ∈ G, |B ∩H| ≤ 1 and |B ∩G| ≤ 1,

2. every two points in V which are not both elements of H belong either to a group or

to exactly one block, but not to both.

A K − IGDD of type (g, h)u is an IGDD which has u groups of size g, each of which

intersects the hole in h points, and K is the set of its block sizes.

Theorem 2.13. [52] A 4 − IGDD of type (g, h)u exists if and only if g ≥ 3h, g(u −

1) ≡ 0 (mod 3), (g − h)(u − 1) ≡ 0 (mod 3), and (g2 − h2)u(u − 1) ≡ 0 (mod 12),

except when (g, h, u) ∈ {(2, 0, 4), (6, 0, 4), (6, 1, 4)}, and possibly except when (g, h, u) ∈

{(15, 3, 14), (21, 3, 14), (93, 27, 14), (15, 3, 18), (21, 3, 18), (93, 27, 18)}.

Let K be a set of positive integers, and let λ be a positive integer. A holey group

divisible design K −HGDD of type (n, hu) is a quadruple (V,G,H,B), where V is a set

of nhu elements, G is a partition of V into n sets of size hu, called groups, H is another

partition of V into u sets of size nh, called holes, and B is a collection of subsets of V ,

called blocks, such that:

1. for every B ∈ B, |B| ∈ K,


2. for every G ∈ G, H ∈ H, and B ∈ B, |G ∩H| = h, |G ∩B| ≤ 1, and |H ∩B| ≤ 1,

3. any pair of points in V which is not contained in any hole H ∈ H is either a subset

of a group G ∈ G or is contained in exactly λ blocks in B, but not both.

For example, we can easily construct a HGDD from a resolvable BIBD(16, 4, 1).

Take the blocks of one parallel class to be the groups. Deleting the blocks of another

parallel class, we obtain four holes, and each of them intersects each group in a point,

that is we get a (4, 1)−HGDD of type (4, 14).

Theorem 2.14. [53] There exists a (3, λ)−HGDD of type (n, hu) if and only if n, u ≥ 3,

λ(u− 1)(n− 1)h ≡ 0 (mod 2), and λu(u− 1)n(n− 1)h2 ≡ 0 (mod 3).

Theorem 2.15. [24] A (4, λ)−HGDD of type (n, hu) exists if and only if n, u ≥ 4 and

λ(u− 1)(n− 1)h ≡ 0 (mod 3), except for (n, h, u) = (4, 1, 6), λ = 1, and possibly except

for λ = 1, h = 3 and (n, u) ∈ {(6, 14), (6, 15), (6, 18), (6, 23)}.

In a HGDD, all holes have the same size. If we allow holes to have different sizes,

we get a double group divisible design. A double group divisible design, DGDD, is a

quadruple (V,H,G,B), where V is a set of points, G and H are two partitions of V into

subsets called groups and holes, respectively, and B is a set of subsets of V , called blocks,

such that

1. for each block B ∈ B, and every G ∈ G and H ∈ H, |B ∩G| ≤ 1 and |B ∩H| ≤ 1,

2. any pair of distinct points from V which is not contained in a hole H ∈ H, is

contained either in a group G ∈ G or in exactly one block B ∈ B, but not in both.

A specific DGDD is a K −DGDD of type (n, hu11 hu22 . . . huss ), where K is the set of

block sizes. It has n groups of size∑s

i=1 hiui, ui holes with nhi points, and each group

intersects ui holes in hi points for all i ∈ {1, 2, . . . , s}. Note that if the set K contains

only one element k, then we denote such a design by k − DGDD. Analogous to the


notation for the GDDs, if we specify the index λ, then admissible pairs of points are

contained in exactly λ blocks.

We construct necessary DGDD used in constructions of GDCDs in Chapters 6 and 7

using the following method.

Construction 2.16. [28] If there exists a K −GDD of type gu11 gu22 . . . guss , and if for all

k ∈ K, there exists a κ−HGDD of type (v, 1k), then there exists a κ−DGDD of type

(v, gu11 gu22 . . . guss ).

Proof. Let (V,G,B) be the K − GDD of type gu11 gu22 . . . guss . Let V × Zv be the set of

elements of the DGDD, where {V × {i} : i ∈ Zv} is its partition into groups, and

{G× Zv : G ∈ G} is its partition into holes. Take the union of blocks of the κ−HGDD

of type (v, 1k) on B × Zv elements for all B ∈ B, |B| = k, to get the blocks of the desired

DGDD.

2.3 Coverings and Packings

Covering and packing generalizations of t-designs have been studied since the 1960’s.

A t − (v, k, λ) covering is a pair (V,B) where V is a v-set, and B is a collection of

k-subsets of V , called blocks, such that any t-subset of V is contained in at least λ distinct

blocks in B. The cover number C(v, k, t) is the minimum possible number of blocks in

a t − (v, k, 1) covering [14]. Note that in the definition of the cover number, the third

argument is the strength and it is assumed that index λ = 1.

A t− (v, k, λ) covering is equivalent to a CARL with strength t, v columns, row limit

k, alphabet size 1, and index λ. We discuss this equivalence relation in Chapter 3.

Optimal size of coverings have been determined for many families. In Chapter 5, we

construct CARLs with a fraction of non-empty cells per row using the optimal t− (v, k, 1)

coverings with t ∈ {2, 3} and c1 <vk≤ c2, where c1, c2 ≥ 1 are constants. In Chapter 6,


we use optimal 2− (v, 4, 1) coverings in our constructions of CARLs with row limit four.

Here we state known results about the cover number for these families.

Theorem 2.17. [38, 39] Let v ≥ 4 be an integer.

C(v, 4, 2) =⌈v4

⌈v−13

⌉⌉+ δ, where δ =

1, v = 7, 9, 10

2, v = 19

0, otherwise.

Theorem 2.18. [14, Thm VI.11.31]

1. C(v, k, 2) = 3 for 1 < v/k ≤ 3/2;

2. C(v, k, 2) = 4 for 3/2 < v/k ≤ 5/3;

3. C(v, k, 2) = 5 for 5/3 < v/k ≤ 9/5;

4. C(v, k, 2) = 6 for 9/5 < v/k ≤ 2;

5. C(v, k, 2) = 7 for 2 < v/k ≤ 7/3, except when 3v = 7k − 1;

6. C(v, k, 2) = 8 for 7/3 < v/k ≤ 12/5, except when 12k − 5v ∈ {0, 1} and v − k is

odd;

7. C(v, k, 2) = 9 for 12/5 < v/k ≤ 5/2, except when 2v = 5k and v − k is odd;

8. C(v, k, 2) = 10 for 5/2 < v/k ≤ 8/3, except when 8k − 3v ∈ {0, 1}, v − k is odd,

and k ≥ 2;

9. C(v, k, 2) = 11 for 8/3 < v/k ≤ 14/5, except when 14k − 5v ∈ {0, 1}, v − k is odd,

and k ≥ 4;

10. C(v, k, 2) = 12 for 14/5 < v/k ≤ 3, except when v = 3k, k 6≡ 0 (mod 3), and k 6≡ 0

(mod 4);

11. C(v, k, 2) = 13 for 3 < v/k ≤ 13/4, except for:

(a) C(13s+ 2, 4s+ 1, 2) = 14, s ≥ 2,

(b) C(13s+ 3, 4s+ 1, 2) = 14, s ≥ 2,

(c) C(13s+ 6, 4s+ 2, 2) = 14, s ≥ 2,

(d) C(19, 6, 2) = 15,


(e) C(16, 5, 2) = 15.

Theorem 2.19. [14, Thm VI.11.34]

1. C(v, k, 3) = 4 for 1 < v/k ≤ 4/3;

2. C(v, k, 3) = 5 for 4/3 < v/k ≤ 7/5;

3. C(v, k, 3) = 6 for 7/5 < v/k ≤ 3/2, except when 2v = 3k and v is odd;

4. C(v, k, 3) = 7 for 3/2 < v/k ≤ 17/11, except when 11v = 17k − 1;

5. C(v, k, 3) = 8 for 17/11 < v/k ≤ 8/5, except when 5v = 8k − 1 and k ≥ 7.

In the definition on a t-covering, if we replace “at least” with “at most,” we get the

definition of a packing. A t− (v, k, λ) packing is a pair (V,B), where V is a v-set, and B

is a collection of k-subsets of V , called blocks, such that every t-subset of V is contained

in at most λ blocks in B . A packing number D(v, k, t) is the maximum number of blocks

in a t− (v, k, 1) packing.

Theorem 2.20. [27, 40] Let v ≥ 3 be an integer.

D(v, 3, 2) =⌊v3

⌊(v−1)

2

⌋⌋− δ, where δ =

1, v ≡ 5 (mod 6)

0, otherwise.

2.3.1 Covering Arrays

We also need some results on covering arrays. First we must recall the notation for

covering arrays.

Definition 2.3. A covering array CAλ(N ; t, k, v) is anN×k array which has k components

represented by the columns of the array. The component i takes values from a v-set

called the alphabet of the component. Moreover, for every {i1, i2, . . . it} ⊂ {1, 2, . . . k},

the N × t subarray obtained by taking columns i1, i2, . . . , it contains each of the distinct

vt t-tuples in these columns at least λ times.


If λ = 1 it is often omitted from the notation. If the number of rows N is not essential

in the context, it is omitted from the notation, as well.

Definition 2.4. The cover number CANλ(t, k, v) of a covering array is the minimum

number of rows N such that a CAλ(N ; t, k, v) exists. An optimal covering array is a

covering array with the minimum size.

We recall three results regarding covering arrays. The first two theorems stated here

take a new perspective on the question of an optimal covering array. Namely, given that

an array has N rows, they consider the question: “What is the maximum number of

columns, so that the array satisfies all the properties of a covering array?”

Theorem 2.21. [21] Let v and N be positive integers, and let k(N, v) be the greatest

positive integer such that a CA(N ; 2, k(N, v), v) exists. Then

lim supN→∞

1

N − vlog2 k(N, v) =

2

v.

Theorem 2.22. [30, 31] Let t = v = 2, let N be a positive integer, and let k(N, 2) be the

greatest positive integer such that a CA(N ; 2, k(N, 2), 2) exists. Then,

k(N, 2) =

(N − 1

bN2c − 1

).

Theorem 2.23. [25] Let t, k and v be positive integers such that t ≤ k. Then,

CAN(t, k, v) ≤ UBca(t, k, v), where

UBca(t, k, v) =

⌈− ln

(e t vt

(kt−1

))ln(1− 1

vt

) ⌉.

An orthogonal array OAλ(t, k, v) is a CAλ(t, k, v) in which every t-tuple is covered

exactly once.


2.4 Probabilistic Methods

To derive the upper bounds on the size of CARLs in Chapter 4, we use the probabilistic

methods listed here. For a general reference on probabilistic methods see [2]. In our work

on approximations, we will frequently use the following well-known inequality:

(2.1)(nk

)k≤(n

k

)<(enk

)k.

If A is an event, P (A) denotes the probability that the event A does not occur.

Theorem 2.24 (Chebyshev’s Inequality). [2] Let X be a random variable and let ε > 0.

Then,

P (|X − E(X)| ≥ εE(X)) ≤ V ar(X)

ε2E(X)2.

Theorem 2.25 (Markov’s Inequality). [2] Given any random variable X in a probability

space, and a constant c,

P (X ≥ c) ≤ E[X]

c.

We also use the main theorem of Rodl’s nibble as presented in [2]. Rodl’s nibble is

one of several variations on the original proof by Rodl [46]. It considers the existence of

an edge cover of a hypergraph. An r-uniform hypergraph H = (V,E) has a set of vertices

V , and a set of edges E, such that each edge is an r subset of V . The degree d(x) of a

vertex x ∈ V is the number of edges in E which contain x. Similarly, a common degree

d(x, y) of two distinct vertices x, y ∈ V is the number of edges in E which contain both x

and y. An edge cover of a graph is a subset of edges such that every vertex is incident to

at least one edge in the cover.

Theorem 2.26. [2, Theorem 4.7.1]

For every integer r ≥ 2 and real numbers k ≥ 1 and a > 0, there are γ = γ(r, k, a) > 0

and d0 = d0(r, k, a) such that for every n such that D ≥ d0 the following holds.


Every r-uniform hypergraph H = (V,E) on a set V of n vertices in which all vertices

have positive degrees and which satisfies the following conditions:

• for all vertices x ∈ V but at most γn of them, d(x) ∈ D(1± γ),

• for all x ∈ V , d(x) < kD,

• for any two distinct x, y ∈ V , d(x, y) < γD,

contains an edge cover of at most (1 + a)nr

edges.

Chapter 3

Definitions and Examples

In this chapter we define the main objects of our study, covering arrays with row limit,

CARLs, and related structures. We introduce the problem in all three equivalent forms

which offer different tools and perspective to the study: array, set, and graph variation.

Regarding the two central questions of minimum size and construction of optimal objects,

we define the cover number and the excess graph, two properties of CARLs.

We also define the packing version of the problem, that is, packing arrays with row

limit, PARLs, and the related structures. We consider properties of PARLs such as the

packing number and the leave graph as well.

3.1 Covering arrays with row limit

We have already mentioned that there are three equivalent variations of definitions of

CARLs. We state and discuss each of them in the following order: the array definition,

the design definition (set variation), and the graph definition.

The array definition

Definition 3.1. A covering array with row limit CARLλ(N ; t, k, {v1, v2, . . . , vk} : w) is

an N × k array with some empty cells, denoted by ‘−’. It has k components which are

24

Chapter 3. Definitions and Examples 25

represented by the columns of the array. The component i takes values from a vi-set

called the alphabet of the component. In a row, there are exactly w components that have

an assigned value from their alphabets. The remaining cells in the row are empty, that is,

the corresponding components have no assigned value. The parameter w is called the

row limit . Moreover, for every {i1, i2, . . . it} ⊂ {1, 2, . . . k}, the N × t subarray obtained

by taking columns i1, i2, . . . , it contains each of the distinct∏t

j=1 vij t-tuples in these

columns at least λ times.

Following the usual notation for covering arrays, the parameter N is called the size, k

is the degree, t is the strength, and λ is the index. Definition 3.1 extends the most general

definition of a covering array, that is the mixed covering array [16,36,44]. If components

accept different number of levels, one cal always define “dummy” alphabet values to

the components with smaller alphabet sizes in order to equalize the number of alphabet

symbols in all components. Here we only study CARLs for which all components have

alphabets of the same size. We denote such CARLs by CARLλ(N ; t, k, v : w), which

means that every component has alphabet size v, and v is called the order . If λ = 1, it

is usually omitted from the notation. Also if the size of the array is not essential in the

context, we omit it from the notation. In particular, a CARL(t, k, v : w) has λ = 1 and

its size is not specified.

Example 3.1. The following is a CARL(8; 2, 5, 2: 4). It has k = 5 columns and N = 8

rows. Each row has exactly w = 4 non-empty cells which take values from the alphabet

set {0, 1}, and hence v = 2. There are four possible 2-tuples of the alphabet symbols:

00, 01, 10, 11. Any 8× 2 subarray, contains every 2-tuple at least once.


1 2 3 4 5

1 0 0 0 0 −

2 0 0 1 − 0

3 1 0 − 1 1

4 1 − 0 1 0

5 0 1 0 − 1

6 1 1 − 0 0

7 1 − 1 0 1

8 0 1 1 1 −

Given parameters t, k, v and w, we are interested in finding an array of the smallest

possible size such that the array satisfies the requirements of a CARLλ(t, k, v : w). A

covering array is a CARL with w = k (see Definition 2.3). Following the notation for the

optimal size of a covering array, we define CARLN , the optimal size of a CARL.

Definition 3.2. The minimum N such that a CARLλ(N ; t, k, v : w) exists is denoted by

CARLNλ(t, k, v : w), and an array of the minimum size is called optimal .

Analogous to the notation of CARLs, we often omit index λ in the notation of

CARLN if λ = 1. The CARL(8; 2, 5, 2: 4) in Example 3.1 is optimal, and hence

CARLN(2, 5, 2: 4) = 8.

In Chapter 1, we mentioned that the optimal size of a covering array is an interesting

and important, but still open question [11]. Given that a covering array is a CARL with

w = k, the question of the optimal size of a CARL for which the row limit is a specified

positive integer valued function of k, w = w(k), arises. In Chapter 4, we study upper and

lower bounds on CARLN(t, k, v : w) as a function of k, and discuss them for some classes

of functions w = w(k).

The design definition


When we are considering pairwise covering, that is when strength t is 2, we can

equivalently consider a CARL as a generalization of GDDs, which motivates the following

definition of group divisible covering designs.

Definition 3.3. A group divisible covering design, a GDCD for short, with index λ, is a

triple (V,G,B), where V is a set of elements, G is a partition of V into subsets, called

groups, and B is a collection of subsets of V , called blocks, such that

1. for any G ∈ G and any B ∈ B, |G ∩B| ≤ 1, and

2. any pair of elements in V which does not belong to a group is contained in at least

λ blocks.

A (K,λ) − GDCD of type gu11 gu22 . . . guss is a GDCD which has ui groups of size gi,

i ∈ [1, s], and K is the set of block sizes. Note that if every pair of elements is contained

either in a group or in exactly λ blocks, but not both, then a GDCD is called a group

divisible design, GDD. The notation rules for GDDs apply to GDCDs. Specifically, if

K = {k} and λ = 1, we usually write k −GDCD instead of ({k}, 1)−GDCD.

Example 3.2. The CARL(8, 2, 5, 2: 4) in Example 3.1 is equivalent to a 4−GDCD of

type 25, (V,G,B). To see the equivalence, let V = Z5 × Z2 be the set of elements of a

GDCD, and let the groups be defined by Gi = {i} × Z2, i ∈ Z5. The group Gi then

corresponds to the column i of the CARL, for i ∈ Z5. The first coordinate of an element

in V denotes the group/component it belongs to and the second coordinate denotes the

alphabet value of the component. Now we can represent each row of the CARL as a set

of four elements in V . The collection of blocks, B, equivalent to the rows of the CARL

given in Example 3.1 is shown in Figure 3.2.

We get that the triple (V,G,B) is a 4−GDCD of type 25. Indeed, a block contains

at most one element of a group since each non-empty cell in the CARL contains exactly


0 1 2 3 40 0 0 0 −0 0 1 − 01 0 − 1 11 − 0 1 00 1 0 − 11 1 − 0 01 − 1 0 10 1 1 1 −

{{(0, 0), (1, 0), (2, 0), (3, 0)},{(0, 0), (1, 0), (2, 1), (4, 0)},{(0, 1), (1, 0), (3, 1), (4, 1)},{(0, 1), (2, 0), (3, 1), (4, 0)},{(0, 0), (1, 1), (2, 0), (4, 1)},{(0, 1), (1, 1), (3, 0), (4, 0)},{(0, 1), (2, 1), (3, 0), (4, 1)},{(0, 0), (1, 1), (2, 1), (3, 1)}}.

Figure 3.1: The CARL(8; 2, 5, 2: 4) from Example 3.1, and the corresponding set ofblocks of an equivalent 4−GDCD of type 25.

one alphabet symbol. For any two elements (i1, j1) ∈ V and (i2, j2) ∈ V such that i1 6= i2,

there exists a block containing them, since there exists a row which contains the value j1

in the column i1 and the value j2 in the column i2.

In general, given a CARL(2, k, v : w) whose columns are labelled by Zk and having Zv

as the alphabet set, we obtain a w−GDCD of type vk, (V,G,B) by setting V = Zk ×Zv;

each group Gi ∈ G corresponds to the column i of the array, so Gi = {i} × Zv, i ∈ Zk,

and each block B ∈ B corresponds to a row in the CARL, so (i, j) ∈ B ⊆ V if and only

if j ∈ Zv appears in column i of the row that B corresponds to. This process is easily

reversible.

Coverings of strength t = 2 are equivalent to GDCDs with group size g = 1. Analo-

gously to CARLN , the minimum size of a CARL, the minimum size of a GDCD with

λ = 1 is called the cover number [28], which comes from the study of coverings.

Definition 3.4. The cover number is the minimum number of blocks such that a

GDCD exists. The cover number of a k − GDCD of type gu11 gu22 . . . guss , is denoted

by C(k, gu11 gu22 . . . guss ). A GDCD with minimum number of blocks is called optimal .

In particular, CARLN(2, k, v : w) = C(w, vk). In Example 3.2, the 4 − GDCD of

type 25 is optimal, and hence C(4, 25) = 8. The optimal size of GDCDs with blocks size

three and index one is known. We state the cover number of this family of objects here.


Later we determine the value of C(4, gu) up to a constant, with some exceptions (see

Theorem 6.2.)

Theorem 3.1. [28] Let g and u ≥ 3 be positive integers. Then,

C(3, gu) =

⌈gu

3

⌈g(u− 1)

2

⌉⌉.

The results on GDDs listed in Chapter 2 demonstrate that GDDs are usually studied

as a family of combinatorial objects with a given block size and variable numbers of groups

and group sizes. Equivalently, one could study a family of CARLs with a fixed row limit

and a variable number of columns and variable alphabet sizes. However, it would not be

practical to record such CARLs since the rows in the array would contain mostly empty

cells. Furthermore, many construction techniques for such families of objects are adapted

from constructions of GDDs. Therefore, in Chapter 6 we study 4−GDCDs rather than

CARLs with row limit four. Also, in Chapter 7, we take another view on constructions

of optimal 3−GDCDs which transform into optimal GDPDs.

When strength t ≥ 3, one can apply the same equivalence relation between a CARL

and a covering version of a group divisible t-design. However, group divisible t-designs

have not been much explored. To our knowledge there are only two papers which consider

this generalization of GDDs [42,43]. In this dissertation, if the strength is two and the

row limit is a constant, we study GDCDs; otherwise, we consider CARLs.

The graph covering definition

Finally, when t = 2, the problem of finding an optimal CARL or a GDCD can

be formulated as a version of the graph covering problem [1, 8]. A graph G is a pair

(V (G), E(G)) where V (G) is the set of vertices of G, and E(G) is the set of edges of G.

Let (V,G,B) be a GDCD with block size k. The set of elements, V , can be considered as

a vertex set of a graph. Then, the set of groups G defines a partition of the vertex set


V into partite sets and every block B ∈ B determines a complete graph Kk induced on

vertices B ⊂ V , which we denote by HB. The graph (V,⋃B∈B E(HB)) has at least one

edge incident to any pair of vertices in two distinct partite sets and it does not contain an

edge between any two vertices in the same partite set. In other words, it has an underlying

complete multipartite graph which may have multiple copies of some edges. The excess

graph of the given k −GDCD is the graph obtained by deleting one copy of each edge

in (V,⋃B∈B E(HB)). Hence, only multiples of an edge are preserved in the excess graph.

More formally, the excess graph of a GDCD, (V,G,B), is the graph with the vertex set

V and the following set of edges: if a pair of elements {x, y} ⊂ V is contained in r(x, y)

blocks in B, the excess graph contains r(x, y)− 1 multiples of an edge incident to vertices

x and y.

Note that the given equivalence relation between GDCDs and graph coverings assumes

that the index λ = 1. This is the only case which we are going to study. However, the

equivalence relation is easily generalized for λ > 1.

Example 3.3. The excess graph of the 4−GDCD of type 25 in Example 3.2 is equivalent

to 2K1,4, two copies of a star with four rays (see Figure 3.3). For example, the pair of

elements {(0, 0), (0, 1)} is contained in the first and the second block of the GDCD in the

order they are listed in Example 3.2. Hence the excess graph has a single edge incident to

vertices (0, 0) and (0, 1).

(0,0)(0,1)

(1,0)

(1,1)

(2,0)(2,1)(3,0)(3,1)

(4,0)

(4,1)

Figure 3.2: Excess graph of the 4−GDCD of type 25 in Example 3.2.


The excess graph of a GDCD is an important property which we consider in our

constructions in Chapters 6 and 7. When t ≥ 3, the problem of finding an optimal

CARL of strength t can be analogously formulated as a question of finding an optimal

hyper-graph edge covering. In the proof of Theorem 4.7, we present another hyper-graph

edge covering problem equivalent to CARLs in order to derive an upper bound on a

CARL with a constant row limit.

3.2 Packing arrays with row limit

If we replace “at least” with “at most” in the definitions of CARLs and GDCDs, we

get the definitions of packing arrays with row limit, PARLs, and group divisible packing

designs, GDPDs, respectively. A packing always exists: taking zero rows or zero blocks,

we get a packing. The main problems are to determine the maximum number of rows or

blocks and to construct such a packing.

We give the formal definitions of a PARL and a GDPD below. The same rules

of notation which hold for CARLs and GDCDs apply to their corresponding packing

variations.

The array definition

Definition 3.5. A packing array with row limit PARLλ(N ; t, k, {v1, v2, . . . , vk} : w) is

an N × k array with some empty cells, denoted by ‘−’. It has k components which are

represented by the columns of the array. The component i takes values from a vi-set

called the alphabet of the component. In a row, there are exactly w components that have

an assigned value from their alphabets. The remaining cells in the row are empty, that is,

the corresponding components have no assigned value. The parameter w is called the

row limit . Moreover, for every {i1, i2, . . . it} ⊂ {1, 2, . . . k}, the N × t subarray obtained

by taking columns i1, i2, . . . , it contains each of the distinct∏t

j=1 vij t-tuples in these

columns at most λ times.


In Chapter 7, we discuss a PARL with index λ = 1 for which all components have

alphabets of size v denoted by PARL(N ; t, k, v : w). We omit N in the notation if it is

not essential in the context. A packing array PAλ(t, k, v) is a PARLλ(t, k, v : k), that is,

it is a PARL with w = k. If λ = 1, it is often omitted in the notation.

Definition 3.6. The maximum N such that a PARLλ(N ; t, k, v : w) exists is denoted by

PARLNλ(t, k, v : w), and an array of the maximum size is called optimal .

The design definition

The set representation of a PARL when t = 2 is called a group divisible packing

design, GDPD.

Definition 3.7. A group divisible packing design, for short a GDPD, with index λ is a

triple (V,G,B), where V is a set of elements, G is a partition of V into subsets, called

groups, and B is a collection of subsets of V , called blocks, such that

1. for any G ∈ G and any B ∈ B, |G ∩B| ≤ 1,

2. any pair of elements in V which is not a subset of a group, is contained in at most

λ blocks.

A (K,λ) − GDPD of type gu11 gu22 . . . guss is a GDPD which has ui groups of size gi,

i ∈ [1, s], and for which K is the set of block sizes. When λ = 1, the index and the

parentheses are usually omitted from the notation. Also, when K = {k}, the curly

brackets are omitted. For example, in Chapter 7 we consider constructions of optimal

3−GDPDs, that is, GDPDs with the set of block sizes K = {3} and index λ = 1.

Note that a k − GDPD of type 1u is equivalent to a 2 − (u, k, 1) packing (see the

definition on page 20). Consistent with the notation for the maximum size of a packing

and the definition of the cover number of a GDCD, we define the packing number of a

GDPD when λ = 1.


Definition 3.8. The maximum number of blocks such that a k − GDPD of type

gu11 gu22 . . . guss exists is called the packing number and it is denoted by D(k, gu11 g

u22 . . . guss ).

A GDPD with maximum number of blocks is called optimal .

The graph covering definition

Denote by Kgu a complete multipartite graph with u partite sets of size g. An

equivalent formulation of the problem of finding an optimal k −GDPD of type gu is to

find in a complete multipartite graph Kgu a maximum number of subgraphs isomorphic to

a complete graph Kk, such that no edge of Kgu is contained in more than one subgraph.

Then, the leave graph is the graph on the vertex set of Kgu whose edge set consists of the

edges of Kgu which are not covered by any of the subgraphs. More formally, the leave

graph of a GDPD, (V,G,B), with index λ = 1, is a simple graph with the vertex set V

and the edge set consisting of the edges incident to the pairs of elements of V which are

not contained in any block or any group of the GDPD.

The definition of the leave graph of a GDPD can be generalized for index λ > 1, in

which case the leave graph may contain multiple edges. However, we only study the leave

graph of GDPDs with index λ = 1.

Example 3.4. Figure 3.3 gives an example of an optimal PARL(5; 2, 5, 2: 4) which is

equivalent to a 4−GDPD of type 25. The equivalence relation is analogous to the one

given in Example 3.2. Let V = Z5 × Z2, and let Gi = {i} × Z2, i ∈ Z5 be a group.

The first coordinate of an element of the GDPD corresponds to a component of the

PARL, and the second coordinate to the alphabet symbol of the component, giving the

equivalence relation between the elements. In Figure 3.3, we present the collection of

blocks B equivalent to the rows of the PARL. Then, (V, {Gi : i ∈ Z5},B) is a 4−GDPD

of type 25. The figure also shows the leave graph of this 4−GDPD which is isomorphic

to two five-cycles, 2C5.


0 1 2 3 41 1 − 0 00 − 0 0 10 1 1 1 −1 0 1 − 1− 0 0 1 0

{(0, 1), (1, 1), (3, 0), (4, 0)}{(0, 0), (2, 0), (3, 0), (4, 1)}{(0, 0), (1, 1), (2, 1), (3, 1)}{(0, 1), (1, 0), (2, 1), (4, 1)}{(1, 0), (2, 0), (3, 1), (4, 0)}

(0,0) (0,1)

(1,0)

(1,1)

(2,0)(2,1)(3,0)(3,1)

(4,0)

(4,1)

Figure 3.3: An optimal PARL(5; 2, 5, 2: 4), the set of blocks of an equivalent 4−GDPDof type 25, and its leave graph.

Chapter 4

Bounds

In this chapter, we study lower and upper bounds on the size of CARLs and we discuss

implications of these bounds for a number of positive integer valued functions w = w(k).

We start with the lower bounds. First we observe how the size of a CARL relates to the

size of a covering array with the same parameters. Then we derive a new lower bound

which we call the Schonheim lower bound since it is a generalization of a lower bound on

the size of a covering.

We derive two probabilistic upper bounds on the size of any CARL, and we discuss

the conditions under which they are asymptotically equal to the lower bounds. We show

that if w(k) is a constant function, then the Schonheim lower bound is optimal (see

Theorem 4.7), and if w(k) = Θ(k), CARLs have logarithmic growth with respect to

k (see Corollaries 4.1 and 4.5). We also improve one of the probabilistic bounds for

w(k) lnw(k) = o(k). At the end of the chapter, in Section 4.3 we compare the derived

upper bounds.

35

Chapter 4. Bounds 36

4.1 Lower bounds

Since covering arrays do not contain any empty cells, for any positive integer valued

function w = w(k) such that t ≤ w(k) ≤ k,

CAN(t, k, v) ≤ CARLN(t, k, v : w).

Hence, the size of a covering array is a lower bound on the size of a CARL.

Recall, in Theorem 2.21 the question of finding an optimal covering array is considered

from a different perspective: for a given positive integers N and v, k(N, v) is the maximum

positive integer such that a CA(N ; 2, k(N, v), v) exists. We generalize this notion in terms

of CARLs. Let N , v, t ≥ 2 be positive integers and let w = w(k) be a positive integer

valued function of k. Then, k(N ; t, v : w) is the maximum number of columns such that a

CARL(N ; t, k(N ; t, v : w), v : w) exists. We have that

k(N ; t, v : w) ≤ k(N ; 2, v : w) ≤ k(N, v),

since a CARL of strength t is also a CARL of strength t− 1, and filling the empty cells

of a CARL with arbitrary alphabet symbols we get a covering array. Then

1

N − vlog2 k(N ; t, v : w) ≤ 1

N − vlog2 k(N, v), so

lim supN→∞

1

N − vlog2 k(N ; t, v : w) ≤ lim sup

N→∞

1

N − vlog2 k(N, v) =

2

v.

Corollary 4.1.

lim supN→∞

1

N − vlog2 k(N ; t, v : w) ≤ 2

v.

Corollary 4.1 states that for any ε > 0 there exists a positive integer N0 such that for


any N ≥ N0,1

N−v log2 k(N ; t, v : w) ≤ 2v

+ ε, that is

N ≥ v +log2 k(N ; t, v : w)

2v

+ ε≥ v +

log2 k2v

+ ε,

for an optimal CARL(N ; t, k, v : w) since k(N ; t, k, v : w) ≥ k by definition. This means

that the size of a CARL as a function of k is asymptotically bounded below by order of

log k for any row limit w.

We derive another lower bound on CARLN in the following section.

4.1.1 The Schonheim lower bound

Recall that a covering is a CARL whose every component has an alphabet of size one.

Coverings have been extensively studied [14, 41]. In 1964, Schonheim gave a lower

bound for the covering number, the minimum number of blocks in a covering [47]. We

generalize the Schonheim lower bound for coverings to a lower bound on the size of a

CARLλ(t, k, v : w). We also refer to this new lower bound as the Schonheim lower bound,

and denote it by SBλ(t, k, v : w).

Theorem 4.2.

CARLNλ(t, k, v : w) ≥ SBλ(t, k, v : w)

where

SBλ(t, k, v : w) =

⌈vk

w

⌈v(k − 1)

w − 1

⌈· · ·⌈v(k − t+ 2)

w − t+ 2

⌈λv(k − t+ 1)

w − t+ 1

⌉⌉· · ·⌉⌉⌉

.

Proof. Let A be an optimal CARLλ(N ; t, k, v : w). Let V be the alphabet of any column.

Denote by r(xc) the number of rows of A which assign the symbol x ∈ V to the component

c ∈ [1, k]. Denote by x0 the symbol in column c0 which appears in the least number of

rows in A, that is r(xc00 ) = min{r(xc) : c ∈ [1, k], x ∈ V }. Consider the subarray formed

by those rows of A which assign x0 to the component c0. When we delete the column c0


from this subarray, we get a CARLλ(r(xc00 ); t− 1, k − 1, v : w − 1). Hence,

r(xc00 ) ≥ CARLNλ(t− 1, k − 1, v : w − 1).

Moreover, by counting non-empty cells in A in two different ways, we get that

Nw =k∑i=1

∑x∈V

r(xc) ≥ vk r(xc00 ).

Hence,

(4.1) N ≥⌈vk

wCARLNλ(t− 1, k − 1, v : w − 1)

⌉.

We can recursively repeat the above process. When t = 1, one can easily construct a

CARL(N ′; 1, k−t+1, v : w−t+1) of size N ′ =⌈λv(k−t+1w−t+1

)⌉. Iterating the inequality (4.1),

we get the Schonheim lower bound on the size of a covering array with row limit.

From now on, we only consider CARLs with index λ = 1. Consistent with the notation

for CARLs, in this case we omit λ from the notation of the Schonheim lower bound and

we write SB(t, k, v : w).

If t and v are constants, and limk→∞kt

w(k)t log2 k= 0, i.e. w(k) = ω

(k

(log k)1/t

), using

inequality (2.1), we have that

limk→∞

SB(t, k, v : w)

log2 k= lim

k→∞

(kt

)vt(1 + o(1))(wt

)log2 k

< limk→∞

(ekt

)tvt(1 + o(1))(wt

)tlog2 k

= 0.

In particular, this means that if w(k) = Θ(k), the Schonheim lower bound given in

Theorem 4.2 does not have logarithmic growth which is expected by Corollary 4.1. Hence,

it cannot be an optimal bound, in general, for CARLs when w(k) = Θ(k).


4.2 Upper Bounds

Now we consider several upper bounds on the size of a CARL. We use probabilistic

methods to derive them. First we give two general upper bounds on the size of any CARL.

Then we derive an upper bound for CARLs for which w(k) lnw(k) = o(k). Finally, we

give even stronger result for CARLs with a constant row limit. In the last section of this

chapter, we compare the obtained bounds.

4.2.1 Notation

In Subsections 4.2.2 and 4.2.3, we construct CARLs one row at a time in two different

ways. The main idea is the same for both of them: we keep adding rows to the array until

CARL requirements are satisfied, that is till all t-tuples are covered. To formalize this

process, we introduce the following definitions and notation.

Let t and v be given positive integers, let w(k) be a positive integer valued function of

k, and let V be a v-set. When t ≤ w(k) ≤ k, we define the following entities:

• A row, ρ = (a1, a2, . . . , ak), is an 1× k array (vector). A row ρ is an admissible row

for a CARL(t, k, v : w) with alphabet set V , if ρ contains exactly w non-empty cells

each of which contains an element from the set V , that is there exists S ⊂ [1, k],

such that |S| = w, and ai ∈ V if i ∈ S or ai =′ −′, otherwise.

• The set of all admissible rows for a CARL(t, k, v : w) is denoted by R and its size

is denoted by R = |R| =(kw

)vw.

• A t-tuple τ = {bc11 , bc22 , . . . , bctt }, where bi ∈ V , and ci ∈ [1, k] for all i ∈ [1, t], is an

assignment of the alphabet element bi to the column ci for t distinct columns.

• The set of all such t-tuples is denoted by T and its cardinality is denoted by

T = |T | =(kt

)vt.


• We say that a row ρ = (a1, a2, . . . , ak) ∈ R covers a t-tuple τ = {bc11 , bc22 , . . . , bctt } ∈

T , if for all i ∈ [1, t], aci = bi.

• An N × k array A = (aij), i ∈ [1, N ], j ∈ [1, k], covers a t-tuple τ ∈ T if there

exists a row in A which covers τ , that is, there exists i0 ∈ [1, N ], such that the row

ρ = (ai0j), j ∈ [1, k], covers τ . A CARL(t, k, v : w) covers all t-tuples in T .

• Each row in R covers r =(wt

)distinct t-tuples. Each t-tuple in T is covered by

exactly D =(k−tw−t

)vw−t rows in R. Notice that Rr = TD, that is

(4.2)

(k

w

)vw(w

t

)=

(k

t

)vt(k − tw − t

)vw−t.

All notions we have defined here depend on t, w, k, and v, which are usually given in the

context.

A CARL(t, k, v : w) covers all t-tuples in T , where T = |T |. Since each row covers

r-tuples, we get that

CARLN(t, k, v : w) ≥ T

r=

(kt

)(wt

)vt.The Schonheim lower bound given in Theorem 4.2 is stronger than this obvious lower

bound. In the following, we derive three upper bounds on the size of a CARL which are

a multiple of this obvious lower bound.

4.2.2 Uniform Distribution Bound

Here we construct a CARL by uniformly at random picking a row from the set of all

possible rows admissible for a CARL, until all requirements of a CARL are satisfied. We

compute the size of such an array in the proof of the following theorem.

Theorem 4.3. Let t, v, k be positive integers and let w(k) be a positive integer valued


function. When t ≤ w(k) ≤ k,

CARLN(t, k, v : w) ≤ UB0(t, k, v : w)

where

UB0(t, k, v : w) =

− ln

(kt

)vt

ln

(1− (w

t)(kt)vt

) .

Proof. Here we use the notation defined in Section 4.2.1. Let R be the set of all admissible

rows for a CARL(t, k, v : w). Let A be a set of N uniformly at random, independently

chosen rows in R. Denote by τ 6∈ A the event that τ is not covered in A, and by τ 6∈ ρ

the event that the t-tuple τ is not covered by ρ, where ρ is a row in A. Since rows of A

are chosen independently, we have that

P (τ 6∈ A) =∏ρ∈A

P (τ 6∈ ρ) =

(1− D

R

)N.

There are T t-tuples in T , so the probability that there exists a τ ∈ T such that τ 6∈ A

is at most T(1− D

R

)N. If N is such that T

(1− D

R

)N< 1, with positive probability there

exists a CARL(N ; t, k, v : w). Applying the identity (4.2), that is(k−tw−t)vw−t

(kw)vw

= DR

= rT

=

(wt)

(kt)vt

, we get that if

(4.3)

(k

t

)vt(

1− (wt)

(kt)vt

)N< 1,

then there exists a CARL(N ; t, k, v : w). The inequality 4.3 is satisfied for any N such

that

N ≥− ln

(kt

)vt

ln

(1− (w

t)(kt)vt

) .


Since w ≤ k,(wt)

(kt)vt

< 1vt

. If v and t are big enough, we can approximate the value of

the above upper bound using the fact that ln(1− x) = −x+ o(x), as x→ 0:

− ln(kt

)vt

ln

(1− (w

t)(kt)vt

) ≈ (kt)(wt

)vt ln

((k

t

)vt).

Note that we obtain a product of the necessary minimum(kt)

(wt)vt and a logarithm of a

function which depends on k. Therefore, if w = Θ(k), we get that the upper bound has

logarithmic growth. We elaborate on asymptotic growth after we derive an improvement

on UB0.

4.2.3 Binomial Bound

We can strengthen the upper bound in Theorem 4.3 if we use a binomial distribution to

sample rows for a CARL.

Theorem 4.4. Let c1 > 1 and c2 > 1 be such that 1c1

+ 1c2< 1. Also, let t and v be

positive integers, and let w(k) be an integer valued function. When t ≤ w(k) ≤ k,


where

UB1(t, k, v : w) =

(kt

)(wt

)vtc1 ln

(ec2c1

(w

t

)).

Proof. Let A be a two dimensional array such that in the set of all admissible rows R for

a CARL(t, k, v : w), any row in R is a row of A with probability

p =ln(c2c1

(wt

))(k−tw−t

)vw−t

=ln c2r

c1

D.

Let X be the number of rows of A. Then X is a random variable with a binomial


distribution B(R, p). Using equality (4.2), we have that

E(X) = Rp =TD

r

ln c2rc1

D=T ln c2r

c1

r.

By Markov’s inequality, P (X ≥ c1E(X)) ≤ 1c1. Hence, P (X < c1E(X)) > 1− 1

c1.

On the other hand, let Y be the number of t-tuples in T not covered by A. Let Iτ ,

τ ∈ T be the indicator variable,

Iτ =

1, τ is not covered by A,

0, τ is covered by A.

Then Y =∑

τ∈T Iτ and

E(Y ) =∑τ∈T

E(Iτ ) =∑τ∈T

P (Iτ = 1) = T (1− p)D ≤ Te−pD = Tc1c2r

,

since 1− p ≤ e−p for all p ∈ R. Again, by Markov’s inequality, P (Y ≥ c2E(Y )) ≤ 1c2

.

Now, if we have two events A and B, P (A ∩B) ≥ P (A)− P (B). By this inequality,

probability that both X < c1E(X) and Y < c2E(Y ) is greater than 1− 1c1− 1

c2> 0 since

1c1

+ 1c2< 1.

Adding a row to cover each not yet covered t-tuple we get a CARL. Hence, with

positive probability

CARLN(t, k, v : w) ≤ c1E(X) + c2E(Y ) ≤ c1T

rlnc2r

c1+ c2T

c1c2r

=T

rc1

(lnc2r

c1+ 1

)=

(kt

)(wt

)vtc1(1 + ln

(c2c1

(w

t

)))=

(kt

)(wt

)vtc1 ln

(ec2c1

(w

t

)).


If w(k) = Θ(k), then there exist 0 < M < 1 and k0 > 0 such that for all k ≥ k0,

Mk ≤ w(k) ≤ k. Then, when t and v are constants, Theorem 4.4 implies that

CARLN(t, k, v : w) ≤(kt

)(wt

)vtc1 ln

(ec2c1

(w

t

))(4.4)

<

(ekt

)t(Mkt

)tvtc1 ln

(ec2c1

(ek

t

)t)= O(ln k).(4.5)

Corollary 4.5. If w(k) ∈ Θ(k) and t ≤ w(k) ≤ k,

CARLN (t, k, v : w(k)) = O(log k).

As we discussed at the beginning of this chapter, CARLN is asymptotically bounded

below by order of log2 k as a consequence of Corollary 4.1. In particular, when w(k) ∈ Θ(k),

the size of a CARL is bounded both below and above by order of log(k), which is a

characteristic of covering arrays. Note that in this case the Schonheim lower bound does

not have logarithmic growth and hence cannot be asymptotically optimal.

More generally, the Schonheim lower bound does not have logarithmic growth if

w(k) = ω(

k(log k)1/t

). If we assume that w(k) = ω

(k

(log k)1/t

)but not necessarily bounded

below by order of k, CARLN is asymptotically bounded below by order of log(k) and

above by log2(k), which is implied by Corollary 4.1 and Theorem 4.4, respectively.

Corollary 4.5 is also a consequence of Theorem 4.3, but the proof requires more steps.

The upper bound UB1 is generally an improvement to UB0 which we show in Section 4.3.

After, in Sections 5.2 and 5.3.2, we compare the size of some constructed families of

CARLs with UB1. Now we approximate the values of constants c1 and c2 in order to

minimize the value of UB1. We use the following approximation of the constants c1 and

c2 as functions of t and w in all comparisons in the coming sections.

As 1c1

+ 1c2< 1, c2 >

c1c1−1 = 1 + 1

c1−1 . Substituting c2 = c1c1−1 in UB1 we get a function


of c1,

f(c1) =

(kt

)(wt

)vtc1(1 + ln

(1

c1 − 1

(w

t

))).

Let F =(kt)

(wt)vt and b = ln

(wt

). Then,

f(c1) = Fc1(1 + b− ln(c1 − 1))

f ′(c1) = F ((1 + b− ln(c1 − 1)) + c1(−1

c1 − 1)).

Hence, f ′(c1) = 0 when g(c1) = b− 1c1−1 − ln(c1 − 1) = 0.

To get the roots of g(c1), we approximate the logarithm. Since 2− c1 < 1, ln(c1− 1) =

ln(1− (2− c1)) ≈ −(2− c1) = c1 − 2, hence

g(c1) ≈ b+ 2− c1 −1

c1 − 1

= −c21 − (b+ 3)c1 + (b+ 3)

c1 − 1.

We get that f(c1) attains the minimum for c1(t, w) ≈ (b+3)−√

(b−1)(b+3)

2> 1. From now on,

we compute the UB1 with the following values for c1 and c2:

c1(t, w) =(ln(wt

)+ 3)−

√(ln(wt

)− 1)(ln

(wt

)+ 3)

2,

c2(t, w) = 1.1 +1

c1(t, w)− 1.

Note that when t is a constant, limw→∞ c1(t, w) = 1. We plot the graphs of the

functions c1(2, w) and c2(2, w) in Figure 4.2.3.


c_1

c_2

6 7 8 9 10w

2

3

4

5

Figure 4.1: Approximation of constants c1 and c2 as functions of w when t = 2.

Second Moment Bound for CARLs when w(k) lnw(k) = o(k)

Here we strengthen the upper bound UB1 when w(k) lnw(k) = o(k) by redoing the proof

of Theorem 4.4 but now we use the second moment instead of Markov’s inequality. Beside

getting a stronger bound, we also get a constructive proof.

First, we introduce some more notation. Let t and v be positive integers, and let

w(k) be a positive integer valued function of k. In addition to the notation established in

Section 4.2.1, we define a common degree of two t-tuples. Assume that t ≤ w(k) ≤ k and

let τ1, τ2 ∈ T . Let d(τ1, τ2) be the number of rows in R which cover both t-tuples τ1 and

τ2. Denote by dmax = max{d(τ1, τ2) : τ1, τ2 ∈ T } =(k−t−1w−t−1

)vw−t−1.

Theorem 4.6. Let t and v be positive integers, and w(k) be an integer valued function.

When limk→∞w(k) lnw(k)

k= 0,


where

UB2(t, k, v : w) =

(kt

)vt(

wt

) (1 + ln

(w

t

))(1 + o(1)) as k →∞.

Proof. Let R be the set of all admissible rows for a CARL(t, k, v : w(k)). Let A be a two


dimensional array such that each row in R is a row of A with probability

p(k) =ln(wt

)(k−tw−t

)vw−t

=ln r

D> 0.

If limk→∞w(k) =∞, then limk→∞ p(k) < limw→∞ln ( ew

t )t

vw−t = 0 by inequality (2.1). On

the other hand, if there exists a positive integer M such that t ≤ w(k) < M for all k, then

for k ≥M , we have that w(k) < k and(k−tw−t

)≥ k−t, so limk→∞ p(k) < limk→∞

ln (Mt )

k−t = 0.

We conclude that in either case, limk→∞ p(k) = 0, so limk→∞(1− p(k))−1/p(k) = e.

Let X be the number of rows of A. Then X is a random variable with a binomial

distribution B(R, p), and

E(X) = Rp =TD

r

ln r

D=T ln r

r=

(kt

)(wt

)vt ln

(w

t

),

and V ar(X) = Rp(1− p). Therefore, using equality (4.2), that is Rr = TD, we get that

V ar(X)

E(X)2=

1− pRp

<1

Rp=

r

TD

D

ln r<r

T=

(wt

)(kt

)vt.

Since, limk→∞w(k)k≤ limk→∞

w(k) lnw(k)k

= 0, for any ε > 0, there exists k1, such that for

all k ≥ k1,V ar(X)ε2E(X)2

<( ew

t )t

( kt )

t < 0.01, for example. Hence, by the second moment method,

with probability at least 0.99, X = E(X)(1± ε) = Tr

ln r(1± ε), when k ≥ k1.

To complete the construction of a CARL(t, k, v : w), we need to add rows to A to

cover the remaining t-tuples. Let Y be the number of t-tuples which have not yet been

covered in A. Let Iτ , τ ∈ T , be the indicator variable,

Iτ =

1, τ is not covered by A,

0, τ is covered by A.

Then Y =∑

τ∈T Iτ . We use the second moment method to determine the value of Y . We


have that, E(Y ) =∑

τ∈T E(Iτ ) =∑

τ∈T P (Iτ = 1) = T (1−p)D. Let µ = E(Iτ ) = (1−p)D.

On the other hand,

V ar(Y ) =∑τ∈T

V ar(Iτ ) + 2∑τ1 6=τ2

Cov(Iτ1 , Iτ2)

= T (µ− µ2) + 2∑τ1 6=τ2

(E(Iτ1Iτ2)− E(Iτ1)E(Iτ2))

= T (µ− µ2) + 2∑τ1 6=τ2

((1− p)2D−d(τ1,τ2) − µ2)

≤ Tµ+ 2∑τ1 6=τ2

µ2((1− p)−d(τ1,τ2) − 1).

Since d(τ1, τ2) ≤ dmax =(k−t−1w−t−1

)vw−t−1 = w−t

(k−t)vD, it follows that

V ar(Y )

E(Y )2≤ Tµ

T 2µ2+T (T − 1)

T 2((1− p)−

w−t(k−t)v

D − 1).

Since 1− p ≤ e−p for any p ∈ R, we have that

(1− p)−w−t

(k−t)vD − 1 ≤ ep

w−t(k−t)v

D − 1 = e(w−t) ln(w

t)(k−t)v − 1.

Now, by the inequality (2.1), 0 <(w−t) ln (w

t)(k−t)v < (w−t)

(k−t)v ln(ewt

)tand since t and v are

constants and limk→∞w lnwk

= 0,

limk→∞

e(w−t) ln(w

t)(k−t)v − 1 = 0.

Also,

limk→∞

Tµ = limk→∞

(k

t

)vt(1− p)

−1p(−pD) = lim

k→∞

(k

t

)vte− ln (w

t) = limk→∞

(kt

)vt(

wt

) = limk→∞

kt

wt=∞.


Therefore, for any ε > 0 , there exists k2 such that for all k ≥ k2,V ar(Y )ε2E(Y )2

< 0.01,

for example, that is, with probability at least 0.99, Y = E(Y )(1 ± ε) = Tµ(1 ± ε) ≤

Te−pD(1± ε) = Te− ln r(1± ε) =(kt)vt

(wt)

(1± ε) = Tr(1± ε).

Each not yet covered t-tuple can be covered with an additional row, and we need to

add at most Y new rows to A. Therefore, for any k ≥ k0, where k0 = max{k1, k2},

CARLN(t, k, v : w) ≤ Y +X =T

r(1 + ln r) (1± ε),

that is

CARLN(t, k, v : w) ≤(kt

)vt(

wt

) (1 + ln

(w

t

))(1± ε).

The proof of Theorem 4.6 is constructive. It outlines a procedure which with high

probability constructs a CARL with size smaller than or equal to UB2. However, we have

the requirement that limk→∞w(k) lnw(k)

k= 0, that is w(k) lnw(k) = o(k). This constraint

comes from the application of the second moment method and the value of the maximum

common degree of any two t-tuples.

Asymptotic size of CARLs when row limit is a constant

When v = 1, Theorem 4.2 is the Schonheim lower bound on the size of covering designs.

Rodl [46] proved that this bound is asymptotically optimal for covering designs. When

w(k) is a constant function, Rodl’s nibble (cf. Theorem 2.26) is directly applicable to

CARLs. We state this result here because the proof of Theorem 2.26 recursively uses

the sampling process presented in the proof of Theorem 4.6. Hence, we get further

improvement of the bound UB2. The recursive proof samples an edge cover of an r-

uniform hypergraph, and it depends on r being a constant. Below we formulate a question

of constructing a CARL in terms of finding an edge cover of an r-uniform hypergraph,

where r =(wt

). Therefore, the row limit w has to be a constant in order r to be a constant.


A CARL(t, k, v : w) is equivalent to an edge cover of a(wt

)-uniform hypergraph. Indeed,

let the set of vertices V of the hypergraph be T , the set of all t-tuples covered by a

CARL(t, k, v : w), and take the set of all possible rows R which are admissible for the

CARL to be the set of edges. We apply Theorem 2.26 to this hypergraph. In the notation

of Theorem 2.26, n = |V | =(kt

)vt. Each row (edge) covers r =

(wt

)t-tuples, so we get an

r-uniform hypergraph, where r is a constant since w is a constant. The degree of any vertex

τ ∈ V is equal to the number of rows in R covering the t-tuple τ , that is D =(k−tw−t

)vw−t.

For any two vertices τ1, τ2 ∈ V , d(τ1, τ2) ≤(k−t−1w−t−1

)vw−t−1 = w−t

(k−t)vD = o(D) since k goes

to infinity and w is a constant. Therefore, Theorem 2.26 states that for any ε > 0, there

exists k0 such that for all k ≥ k0 such a hypergraph has an edge cover of size at most

|V |r

(1 + ε). An edge cover is equivalent to a subset of rows in R such that each t-tuple

is contained in at least one row; it is a CARL. Hence, if t, v and w are constants, then

CARLN(t, k, v : w) ≤ (kt)

(wt)vt(1 + o(1)) when k →∞.

On the other hand, Theorem 4.2 gives the Schonheim lower bound on the size of a

CARL(t, k, v : w) which is

SB(t, k, v : w) =

⌈vk

w

⌈v(k − 1)

w − 1

⌈· · ·⌈v(k − t+ 2)

w − t+ 2

⌈v(k − t+ 1)

w − t+ 1

⌉⌉· · ·⌉⌉⌉

=

(kt

)(wt

)vt(1 + o(1)).

Therefore, the Schonheim lower bound is asymptotically equal to the upper bound.

We summarize these results in the following theorem.

Theorem 4.7. Given positive integers t ≤ w and v, let k be a positive integer such that

t ≤ w ≤ k. Then,

CARLN(t, k, v : w) =

(kt

)(wt

)vt(1 + o(1)) as k →∞.

Analogous to the proof of Theorem 4.6, the proof of Theorem 2.26 is constructive.


Hence, we can construct such a CARL with high probability. Theorems 3.1 and 6.2

strengthen the result of Theorem 4.7 when w = 3 or 4 and t = 2. Moreover, when t = 2,

constructing an optimal CARL is equivalent to finding an optimal edge covering of a

complete multipartite graph with copies of a complete graph. Alon, Caro and Yuster [1]

studied a more general graph covering problem. They gave an equivalent bound to the

Schonheim lower bound in Theorem 4.2 for t = 2, and proved that for a fixed w and

sufficiently large number of components k an optimal covering exists. However, their proof

is based on the unpublished work of Gustavsson. We have shown how Rodl’s nibble gives

an asymptotic solution for this particular graph covering problem which is independent

of Gustavsson’s work.

4.3 Bounds comparison

In this chapter, we have derived two probabilistic bounds on the size of any CARL, UB0

and UB1 in Theorems 4.3 and 4.4, respectively. Further, we strengthen UB1 for two

classes of functions w = w(k) (see Theorems 4.6 and 4.7). We list the obtained results

here and discuss how they compare to one another.

Let parameters t and v be positive integers, w = w(k) be a positive integer valued

function such that t ≤ w(k) ≤ k, c1, c2 ≥ 1 such that 1c1

+ 1c2< 1. Then, we have that:

UB0(t, k, v : w) =

− ln

(kt

)vt

ln

(1− (w

t)(kt)vt

)

UB1(t, k, v : w) =

(kt

)(wt

)vtc1 ln

(ec2c1

(w

t

))=

(kt

)(wt

)vtc1(1 + ln

(c2c1

(w

t

)))UB2(t, k, v : w) =

(kt

)vt(

wt

) (1 + ln

(w

t

))(1 + o(1)) when w(k) lnw(k) = o(k)


We start by comparing UB0 and UB1.

UB1(t, k, v : w)

UB0(t, k, v : w)≤

(kt)

(wt)vtc1 ln

(c2ec1

(wt

))− ln (k

t)vt

ln

(1−

(wt)

(kt)vt

) = −(kt

)(wt

)vt ln

(1−

(wt

)(kt

)vt

)c1

ln(c2ec1

(wt

))ln(kt

)vt

.

Now, − (kt)

(wt)vt ln

(1− (w

t)(kt)vt

)= 1 + o(1) if

(wt)

(kt)vt

is sufficiently close to 0. In particular

if w(k) = o(k), we get that

limk→∞

UB1(t, k, v : w)

UB0(t, k, v : w)= 0,

so UB1(t, k, v : w) = o(UB0(t, k, v : w)). If w(k) = Θ(k), we already mentioned that both

UB0 and UB1 have the same rate of growth, which we can see here as well. The ratio of

UB1 over UB0 is bigger if w(k) is closer to k. Figure 4.2 gives three examples for which

w(k) = 910k. These examples show that UB1 is generally better than UB0. However, one

can see that if t = 2, v = 2 and w(k) = 910k, UB0 is better than UB1. Therefore, there

are cases in which UB0 is better than UB1.

The upper bounds UB1 and UB2 are similar. The constant c1 in UB1 is replaced by

1 + o(1) expression in UB2. However, UB2 does not have(wt

)multiplied by c2

c1> 1, which

gives us an improvement in UB2 compared to UB1.

We can make another comparision: when w(k) = k, UB1 becomes an upper bound on

the size of a covering array, and it can be compared to UBca, given in Theorem 2.23 [25],

UBca is an important upper bound on the size of covering arrays. Using the approximations

of the values of c1 and c2 as functions of t and w given in Section 4.2.3, Figure 4.3 compares

UB1 when w(k) = k and UBca for strength t = 2 and alphabet sizes v ∈ {10, 100}. It

shows that when the alphabet size v = 10, UBca is smaller than UB1. However, if v = 100,

UB1 is smaller than UBca for all k < 850. If we approximate ln(1− 1

vt

)≈ − 1

vt, we


UB_0

UB_1

50 100 150 200k

3000

3500

4000

4500

5000

5500

N

UB_1

UB_0

50 100 150 200k

250

300

350

400

N

UB1(2, k, 10: k/2) , UB0(2, k, 10: k/2) UB1(2, k, 5: 9k/10), UB0(2, k, 5: 9k/10)

UB_0

UB_1

50 100 150 200k

2500

3000

3500

4000

N

UB_0

UB_1

50 100 150 200k

60 000

80 000

100 000

120 000

140 000

160 000N

UB1(2, k, 15: 9k/10) , UB0(2, k, 15: 9k/10) UB1(5, k, 5: 9k/10), UB0(5, k, 5: 9k/10)

Figure 4.2: Comparison of UB1 and UB0.

get that UBca(t, k, v) ≈ vt(1 + ln

(t vt(k−1t−1

)))= vt ln

(et2 vt

k

(kt

)). Then, for w(k) = k,

one can compute that UB1(t, k, v : k) is smaller than or equal to this approximation of

UBca(t, k, v) when

k

(k

t

)c1−1≤ t2 vt

ec1−1

(c1c2

)c1.

When the inequality holds, UB1 is better than this important upper bound on the size of

covering arrays.


UB_1

UB_ca

200 400 600 800 1000k

1000

1200

1400

1600

1800

N

UB_1

UB_ca

200 400 600 800 1000k

120 000

130 000

140 000

150 000

160 000

170 000

180 000

N

UB1(2, k, 10: k) , UBca(2, k, 10) UB1(2, k, 100: k), UBca(2, k, 100)

Figure 4.3: Comparison of UB1 and UBca when t = 2.

Chapter 5

General Constructions

In this chapter we construct families of CARLs whose size is in between the lower and

upper bounds derived in the previous chapter. First, we give an algorithm for construction

of a CARL whose the size is at most equal to UB0. The algorithm constructs an array

one row at a time, always covering at least the average number of not yet covered t-tuples

in a row. However, we have already shown that UB1 is generally a better bound than UB0.

Also, the algorithm may run in exponential time depending on the function w = w(k).

We also study two combinatorial constructions of CARLs which we apply to get a

number of families of CARLs with a constant ratio kw(k)

. The sizes of these families are

considerably smaller than predicted by the upper bounds UB0 and UB1.

5.1 Algorithmic construction

Here we give an algorithm for construction of CARLs of the size at most equal to UB0

given in Theorem 4.3. Automatic Efficient Test Generator, AETG [9], is an algorithm

for construction of covering arrays. Analogous to AETG, there is a greedy algorithm for

construction of CARLs. It relies on choosing one row at a time which covers at least

the average number of the t-tuples which are not already covered. First we compute this

average in Lemma 5.1. Then we use this lemma to construct a CARL and compute its

55

Chapter 5. General Constructions 56

size. In the following we use the notation defined in Section 4.2.1.

Lemma 5.1. Let t and v be positive integers, and let w(k) be an integer function. When

t ≤ w(k) ≤ k, assume that A is an M × k subarray of a CARL(N ; t, k, v : w), M ≤ N ,

and let R be the set of all admissible rows for a CARL(t, k, v : w). If A does not cover

α of the t-tuples in T , then there exists a row in R which covers at least(wt)

(kt)vt

α of the

t-tuples which are not covered in A.

Proof. Let

W = {(ρ, τ) : τ is a t− tuple not covered by A, and ρ ∈ R is a row which contains τ}.

Every t-tuple which is not already covered in A is contained in D =(k−tw−t

)vw−t rows in R,

none of which are already in A . Hence,

(5.1) |W | = αD.

On the other hand, every row ρ ∈ R appears in W as many times as there are t-tuples

not covered by A which are covered by ρ. Note that if ρ is already in A, then it covers 0

of the not yet covered t-tuples. For a given ρ ∈ R, denote by νρ the number of t-tuples

not covered by A covered by the row ρ. Recall that R = |R|. We have that

(5.2) |W | =∑ρ∈R

νρ ≤ Rν,

where ν = max{νρ : ρ ∈ R}.

From the equation 5.1 and 5.2, we get that

α

(k − tw − t

)vw−t ≤

(k

w

)vwν,


and by equation (4.2),

ν ≥(k−tw−t

)vw−t(

kw

)vw

α =

(wt

)(kt

)vtα.

Let t, v and k be positive integers and let w = w(k) be a positive integer valued

function such that t ≤ w(k) ≤ k. We construct a CARL(t, k, v : w) one row at a time

using Lemma 5.1. Denote by αn the number of t-tuples which remain to be covered after

the addition of the nth row. To start with, the number of rows in the array is n = 0 and

the number of t-tuples which are not covered is α0 =(kt

)vt. By Lemma 5.1, there exists a

row which covers at least(wt)

(kt)vt(kt

)vt =

(wt

)t-tuples which are not already covered. Hence,

α1 ≤(kt

)vt −

(wt

). Iterating this process n times, each time choosing a row which covers

at least the average number of t-tuples which are not already covered, We get that

αn ≤((

k

t

)vt −

(w

t

))(1−

(wt

)(kt

)vt

)n−1

.

Since a CARL(t, k, v : w) covers all t-tuples, we need αn < 1. Hence, for any n such that

n ≥ 1−ln((kt

)vt −

(wt

))ln

(1− (w

t)(kt)vt

)=− ln

((kt

)vt)

ln

(1− (w

t)(kt)vt

) ,

we get that αn < 1. In other words, the size of the constructed CARL is smaller than or

equal to UB0(t, k, v : w).

The running time of the algorithm depends on choosing an optimal row to append to

the array in each step and this can require as many as(kw

)vw steps. If w is an unbounded

function of k, vw(k) is exponential. However, when w(k) is a constant function, the

algorithm runs in polynomial time of k.


For covering arrays, the idea of constructing an array one row at a time by adding a

row which covers at least the average number of t-tuples which are not already covered is

paired with the idea of ‘filling in one cell at a time’ by a greedy, polynomial time algorithm

called the Deterministic Density Algorithm, DDA for short [6, 13]. For a given row, the

algorithm fills in a random order the cells of the row and it ensures that the constructed

row covers at least the average number of t-tuples which are not already covered. To

get a CARL, an obvious extension of the DDA algorithm would be to terminate a row

construction after w cells have been filled. However, this would not guarantee that we

have covered the average number of t-tuples which are not already covered in the new

row. If we were to adapt the DDA algorithm, we would first need to pick w columns

which contain more than the average number of t-tuples which are not already covered

compared to any choice of w columns. This can be done in linear time if w is a constant

or differs from k by a constant, but if, for example, w(k) = k/2, then(kw

)=(kk/2

)> 2k√

2k.

Our algorithm constructs a CARL for any given set of parameters. However, the size

of the constructed CARL may exceed the bound UB1 (see Section 4.3). In the following

two sections we consider two combinatorial techniques of constructing CARLs. We apply

them to obtain a number of families of CARLs with a constant ratio kw(k)

and the size

smaller than the bounds UB0 and UB1. The first one is the product construction.

5.2 Product construction of CARLs with t = 2

The so called copy-paste or product construction of covering arrays [12] is applicable

for the construction of CARLs with strength t = 2 as well. We adopt this method for

CARLs here, and then apply it to construct several families of CARLs.

Construction 5.2. If a CARL(N1; 2, k1, v : w1) and a CARL(N2; 2, k2, v : w2) exist, then

there exists a CARL(N1 +N2; 2, k1k2, v : w), where w = max{k1w2, k2w1}.

Proof. Let A be a CARL(N1; 2, k1, v : w1) and let B be a CARL(N2; 2, k2, v : w2). Con-


struct an N1× k1k2 array, C1, by appending k2 copies of A. Then construct an N2× k1k2

array, C2, by appending first k1 copies of the first column of B, then k1 copies of the

second column of B, and continuing this process till we attach k1 copies of kth2 column of

B. To finish the construction, align the columns of C1 and C2 on top of each other.

More formally, denote by C(ρ, c) the entry in the row ρ ∈ [1, N1 + N2] and column

c ∈ [1, k1k2] of the desired array C. If ρ ≤ N1, let C(ρ, ik1 + c) = A(ρ, c), where

i ∈ [0, k2 − 1] and c ∈ [1, k1]; if N1 < ρ ≤ N1 +N2, let C(ρ, (c− 1)k1 + i) = B(ρ−N1, c),

for i ∈ [1, k1] and c ∈ [1, k2].

Given the name product construction, we say that an array A is multiplied by an

array B to denote an application of the product construction using the arrays A and B.

The product construction preserves the ratio wk

and logarithmic growth. Indeed, given

a constant 0 < c < 1, if w1(k1) = ck1, and w2(k2) = ck2, Construction 5.2 yields a CARL

with k = k1k2 columns and row limit w = ck1k2, that is, w(k) = ck. If we apply the

product construction on two families of CARLs which have logarithmic growth, then the

resulting CARL has the size N1 + N2 = Θ(log k1) + Θ(log k2) = Θ(log(k1k2)) and k1k2

columns.

We can apply product construction recursively. Given a CARL A, start by multiplying

A by itself. Then, multiply A with the result of the previous iteration. A CARL(2, k, v : w)

is equivalent to a w −GDCD of type vk. Many families of optimal GDCDs with w = 3

or 4 are known. Applying the product construction recursively starting with a 3−GDCD

given in Theorems 3.1 yields the following result.

Theorem 5.3. Let v and k ≥ 3 be positive integers. Then for any positive integer n,

CARLN(2, kn, v : 3kn−1) ≤ n

⌈vk

3

⌈v(k − 1)

2

⌉⌉.

Theorem 4.7 proves that the Schonheim lower bound is asymptotically optimal when w

is a constant. Hence, for any given integers v and k ≥ 4, there exists δ = δ(v, k), such that


a 4−GDCD of type vk has at most SB(2, k, v : 4) + δ blocks, and limk→∞δ(v,k)

SB(2,k,v : 4)= 0.

In Chapter 6 we prove that δ is a constant, independent of v and k, with some possible

exceptions (see Theorem 6.2). Now, we apply the product construction recursively to

4−GDCDs.

Theorem 5.4. Let v and k ≥ 4 be positive integers, and let δ ≥ 0 be such that there

exists a 4−GDCD of type vk which has⌈vk4

⌈v(k−1)

3

⌉⌉+ δ blocks. Then for any positive

integer n,

CARLN(2, kn, v : 4kn−1) ≤ n

⌈vk

4

⌈v(k − 1)

3

⌉⌉+ nδ.

We can now apply the product construction using CARLs constructed in Theo-

rems 5.3 and 5.4. If the ingredients in the product construction have the same ratio of the

number of columns to the row limit, the product construction preserves this ratio as well.

Theorem 5.5. Let v, k1 ≥ 3 and k2 ≥ 4 be positive integers such that 3k2 = 4k1, and let

δ ≥ 0 be such that there exists a 4−GDCD of type vk2 with⌈vk24

⌈v(k2−1)

3

⌉⌉+ δ blocks.

Then for any two positive integers n and m,

CARLN(2, kn1km2 , v : 3kn−11 km2 ) ≤ n

⌈vk13

⌈v(k1 − 1)

2

⌉⌉+m

⌈vk24

⌈v(k2 − 1)

3

⌉⌉+mδ.

Proof. We only compute the value of the row limit. Define An to be the array obtained

by recursively applying the product construction n times, starting with an array A. Let

A1 be an array equivalent to an optimal 3−GDCD of type vk1 , and let A2 be an array

equivalent to a 4 − GDCD of type vk2 . The row limit of A1n is w1 = 3kn−11 . On the

other hand, the row limit of A2m is w2 = 4km−12 . Therefore, the product construction

applied to A1n and A2

m yields an array with w = min{kn1w2, km2 w1}. Since 3k2 = 4k1,

we get that kn1w2 = kn1 4 km−12 = kn−11 3 k2 km−12 = km2 w1.

Figure 5.1 gives four examples which compare CN(n), the size of the constructed


CARLs from Theorems 5.3 and 5.4, against the upper bound UB1 and the Schonheim

lower bound SB for some fixed values of k and v, and w ∈ {3, 4}. The examples in

Figure 5.2 do the same for CARLs constructed in Theorem 5.5 when m = n. All

4−GDCDs used in the examples presented in Figures 5.1 and 5.2 meet the Schonheim

lower bound, i.e. δ = 0 (see Theorem 6.59). One can see that the constructed CARLs

have considerably smaller size than that predicted by UB1.

UB_1

CN

SB

2 4 6 8 10 12n

5000

10 000

15 000

20 000

25 000

N

UB_1

SB

CN

2 4 6 8 10n

5.0 ´ 106

1.0 ´ 107

1.5 ´ 107

2.0 ´ 107

2.5 ´ 107

3.0 ´ 107

N

UB1(2, 12n, 5: 3 · 12n−1), UB1(2, 21n, 101: 3 · 21n−1),CN(n) = nSB(2, 12, 5: 3), CN(n) = nSB(2, 21, 101: 3),SB(2, 12n, 5: 3 · 12n−1) SB(2, 21n, 101: 3 · 21n−1)

UB_1

CN

SB

2 4 6 8 10 12n

2000

4000

6000

8000

10 000

12 000

14 000

N

UB_1

CN

SB

2 4 6 8 10n

2 ´ 106

4 ´ 106

6 ´ 106

8 ´ 106

1 ´ 107

N

UB1(2, 12n, 5: 4 · 12n−1), UB1(2, 17n, 97: 4 · 17n−1),CN(n) = nSB(2, 12, 5: 4), CN(n) = nSB(2, 17, 97: 4),SB(2, 12n, 5: 4 · 12n−1) SB(2, 17n, 97: 4 · 17n−1)

Figure 5.1: Comparison of CN , the size of CARLs constructed in Theorems 5.3 and 5.4,UB1(t, k, v : w) and SB(t, k, v : w).

We do a more formal comparison between the size of the family of CARLs constructed

in Theorem 5.3 and UB1. Similar comparison can be done for Theorems 5.4 and 5.5. Let


UB_1

CN

SB

2 3 4 5n

5000

10 000

15 000

20 000

N

UB_1

CN

SB

2 3 4 5n

10 000

20 000

30 000

40 000

50 000

60 000

70 000

N

UB1(2, 12n16n, 5: 3 · 12n−116n), UB1(2, 15n20n, 7: 3 · 15n−120n),CN(n) = nSB(2, 12, 5: 3)+ CN(n) = nSB(2, 15, 7: 3)

+nSB(2, 16, 6: 4) +nSB(2, 20, 7: 4)SB(2, 12n16n, 5: 3 · 12n−116n) SB(2, 15n20n, 7: 3 · 15n−120n)

Figure 5.2: Comparison of CN , the size of CARLs constructed in Theorems 5.5 whenm = n, UB1(t, k, v : w) and SB(t, k, v : w).

k, v and n be positive integers. Then:

CN(2, kn, v : 3kn−1)

UB1(2, kn, v : 3kn−1)=

n⌈vk3

⌈v(k−1)

2

⌉⌉kn(kn−1)

3kn−1(3kn−1−1)v2c1 ln

(ec2c1

3kn−1(3kn−1−1)2

)<

3n(vk3v(k−1)+2

2+ 1)

k kn−13kn−1−1v

2 ln 3kn−1(3kn−1−1)2

<3n(vk3v(k−1)+2

2+ 1)

k kn−13kn−1−1v

2(2n− 2) ln k

−→n→∞

3(vk3v(k−1)+2

2+ 1)

2k2

3v2 ln k

≤ 3

4

k2 + k + 6

k2 ln k< 1 for k ≥ 9.

5.3 Wilson’s Construction

In this section, we give a design construction for CARLs, which is based on Wilson’s

Fundamental Construction for GDDs (see [54] or [14, (IV 2.5)]). Analogously, we call

this construction method Wilson’s construction for CARLs, and it is applicable for any

strength t.

In the following, given a CARLλ(t, k, v : w), A, denote by A(ρ, c) the entry of A in


the row ρ and the column c. Recall, a t − (v, k, λ) covering with v elements and block

size k is equivalent to a CARLλ(t, v, 1: k).

Construction 5.6. If there exist a t− (κ,K, λ1) covering, (X,B), and a

CARLλ2(Nk; t, k, v : w) for every k ∈ K, then there exists a CARLλ1λ2(N ; t, κ, v : w),

where N =∑

B∈BN|B|.

Proof. We want to construct a CARL(t, |X|, v : w), C. Label the columns of C by the

elements of the set X. Let Ak be a CARLλ2(Nk; t, k, v : w), where k ∈ K. For every

B ∈ B, where |B| = k ∈ K, construct N|B| rows of C, {ρBi: i ∈ {1, 2, . . . , N|B|}}, such

that for x ∈ X,

C(ρBi, x) = A|B|(i, x), if x ∈ B,

C(ρBi, x) = ′−′, if x 6∈ B,

where ′−′ denotes an empty cell. Now, C is a CARLλ1λ2(N ; 2, |X|, v : w) of size N =∑B∈BN|B|. Indeed, if {x1, x2, . . . , xt} ⊂ X, then there are at least λ1 sets Bj ∈ B such

that {x1, x2, . . . , xt} ⊂ Bj for all j ∈ {1, 2, . . . , λ1}. Since, for every j, A|Bj | covers all t-

tuples of elements from the alphabet in any t columns at least λ2 times, C covers all t-tuples

in columns {x1, x2, . . . , xt} at least λ1λ2 times in rows {ρBji

: i ∈ {1, 2, . . . , N|B|}, j ∈

{1, 2, . . . , λ1}}.

5.3.1 Applications of Wilson’s Construction

Here we construct CARLs of strength t ∈ {2, 3} for which c1 < k/w ≤ c2, where c1 > 1

and c2 > 1 are constants. In particular, we apply Wilson’s construction using the coverings

with large block sizes given in Theorems 2.18 and 2.19 and orthogonal arrays or covering

arrays.

Theorem 5.7. Let q be a prime power, and let w ∈ {2, 3, . . . , q + 1}. Then,


1. CARLN(2, k, q : w) ≤ 3q2 for 1 < k/w ≤ 3/2;

2. CARLN(2, k, q : w) ≤ 4q2 for 3/2 < k/w ≤ 5/3;

3. CARLN(2, k, q : w) ≤ 5q2 for 5/3 < k/w ≤ 9/5;

4. CARLN(2, k, q : w) ≤ 6q2 for 9/5 < k/w ≤ 2;

5. CARLN(2, k, q : w) ≤ 7q2 for 2 < k/w ≤ 7/3, except possibly when 3k = 7w − 1;

6. CARLN(2, k, q : w) ≤ 8q2 for 7/3 < k/w ≤ 12/5, except possibly when 12w − 5k ∈

{0, 1} and k − w is odd;

7. CARLN(2, k, q : w) ≤ 9q2 for 12/5 < k/w ≤ 5/2, except possibly when 2k = 5w

and k − w is odd;

8. CARLN(2, k, q : w) ≤ 10q2 for 5/2 < k/w ≤ 8/3, except possibly when 8w − 3k ∈

{0, 1}, k − w is odd;

9. CARLN(2, k, q : w) ≤ 11q2 for 8/3 < k/w ≤ 14/5, except possibly when 14w−5k ∈

{0, 1}, k − w is odd, and w ≥ 4;

10. CARLN(2, k, q : w) ≤ 12q2 for 14/5 < k/w ≤ 3, except possibly when k = 3w,

w 6≡ 0 (mod 3), and w 6≡ 0 (mod 4).

11. CARLN(2, k, q : q) = 13q2 for 3 < k/w ≤ 13/4, except possibly for:

(a) CARLN(2, 13s+ 2, q : 4s+ 1) ≤ 14q2, s ≥ 2,

(b) CARLN(2, 13s+ 3, q : 4s+ 1) ≤ 14q2, s ≥ 2,

(c) CARLN(2, 13s+ 6, q : 4s+ 2) ≤ 14q2, s ≥ 2,

(d) CARLN(2, 19, q : 6) ≤ 15q2,

(e) CARLN(2, 16, q : 5) ≤ 15q2.

Proof. Apply Construction 5.6 on the covering with strength t = 2 and block size w given

in Theorem 2.18 and an orthogonal array OA(2, w, q), which is any q2 ×w subarray of an

OA(2, q + 1, q) given in Theorem 2.12.

Analogously, we apply Wilson’s Construction on the coverings given in Theorem 2.18

and the optimal covering arrays with alphabet size v = 2 and strength t = 2, given in

Theorem 2.22 to obtain the following theorem.


Theorem 5.8. Given a positive integer n ≥ 2, let w(n) ≤(n−1bn2c−1

).

1. CARLN(2, k, 2: w(n)) ≤ 3n for 1 < k/w(n) ≤ 3/2;

2. CARLN(2, k, 2: w(n)) ≤ 4n for 3/2 < k/w(n) ≤ 5/3;

3. CARLN(2, k, 2: w(n)) ≤ 5n for 5/3 < k/w(n) ≤ 9/5;

4. CARLN(2, k, 2: w(n)) ≤ 6n for 9/5 < k/w(n) ≤ 2;

5. CARLN(2, k, 2: w(n)) ≤ 7n for 2 < k/w(n) ≤ 7/3, except possibly when 3k =

7w(n)− 1;

6. CARLN(2, k, 2: w(n)) ≤ 8n for 7/3 < k/w(n) ≤ 12/5, except possibly when

12w(n)− 5k ∈ {0, 1} and k − w(n) is odd;


2k = 5w(n) and k − w(n) is odd;


8w(n)− 3k ∈ {0, 1}, k − w(n) is odd;


14w(n)− 5k ∈ {0, 1}, k − w(n) is odd, and w(n) ≥ 4;

10. CARLN(2, k, 2: w(n)) ≤ 12n for 14/5 < k/w(n) ≤ 3, except possibly when k =

3w(n), w(n) 6≡ 0 (mod 3), and w(n) 6≡ 0 (mod 4).

(a) CARLN(2, 13s+ 2, 2: w(n)) ≤ 14n, w(n) = 4s+ 1, s ≥ 2,

(b) CARLN(2, 13s+ 3, 2: w(n)) ≤ 14n, w(n) = 4s+ 1, s ≥ 2,

(c) CARLN(2, 13s+ 6, 2: w(n)) ≤ 14n, w(n) = 4s+ 2, s ≥ 2,

(d) CARLN(2, 19, 2: 6) ≤ 90,

(e) CARLN(2, 16, 2: 5) ≤ 90.

In the similar manner, we construct several families of CARLs with strength t = 3

and with k/w being in between two fractions.


Theorem 5.9. Let q be a prime power, and let w ∈ {3, 4, . . . , q + δ}, where

δ =

2, when q = 2n, n ≥ 2,

1, otherwise.

Then,

1. CARLN(3, k, q : w) ≤ 4q3 for 1 < k/w ≤ 4/3;

2. CARLN(3, k, q : w) ≤ 5q3 for 4/3 < k/w ≤ 7/5;

3. CARLN(3, k, q : w) ≤ 6q3 for 7/5 < k/w ≤ 3/2, except possibly when 2k = 3w and

k is odd;

4. CARLN(3, k, q : w) ≤ 7q3 for 3/2 < k/w ≤ 17/11, except possibly when 11k =

17w − 1;

5. CARLN(3, k, q : w) ≤ 8q3 for 17/11 < k/w ≤ 8/5, except possibly when 5k = 8w−1

and w ≥ 7.

Proof. Apply Construction 5.6 using the coverings of size t = 3 given in Theorem 2.19

with block size w and an orthogonal array OA(3, w, q), which is any q3 × w subarray of

the orthogonal array given in Theorem 2.12.

5.3.2 Analysis of the constructed objects

In this section, we compare the size of the constructed objects in Theorems 5.7, 5.8, and 5.9

against the Schonheim lower bound (Theorem 4.2), and the upper bound UB1 given

in Theorem 4.4. In Figures 5.3 and 5.4, we denote by CN(t, k, v : w) the size of the

constructed CARL in Theorems 5.7-5.9.

In Theorems 5.7 and 5.9, we have a choice of w ∈ {t, t + 1, . . . , q + 1}. If we let

w(k) = t, it is easy to see that CARLN(t, k, v : t) =(kt

)qt, which is exactly what we

get in Theorems 5.7 and 5.9, unless they are one of the exceptions of the original. The

other extreme case is when w = q + 1, and c1(q + 1) < k ≤ c2(q + 1), for the rational


numbers c1, c2 > 1 given in the theorems. In the examples in Figure 5.3, the size of

CARLs constructed in Theorems 5.7 and 5.9 is considerably smaller than UB1 and is

relatively close to the Schonheim bound. For example, if we compare the ratio of the size

of the constructed CARL in Theorem 5.7, CN(2, k, q : w) = cq2 when a1 <kw≤ a2, q is

a prime power and c is the positive integer given in Theorem 5.7 corresponding to a1 and

a2, and SB(2, k, q : w), we get that

CN(2, k, q : w)

SB(2, k, q : w)=

cq2⌈q kw

⌈q (k−1)w−1

⌉⌉ ≤ cq2⌈q kw

⌈q kw

⌉⌉ ≤ cq2

qkwqkw

=c(kw

) < c

a1.

Hence, the size of the constructed families of CARLs is within a constant multiple of the

Schonheim bound, independent of q. Similar computation can be done for the results of

Theorem 5.9.

The examples in Figure 5.4 illustrate the comparison of the size CN(n) of the CARLs

constructed in Theorem 5.8 when w(n) =(

n−1bn/2c−1

), the Schonheim lower bound, SB, and

the upper bounds UB0 and UB1 for the same parameters. As expected, the Schonheim

bound is constant and does not reflect the expected growth. Also, the size of the

constructed CARLs is considerably smaller than UB0 and UB1. Since v = 2 and t = 2,

we also have that UB0 is smaller than UB1.

More generally, let w(n) =(

n−1bn/2c−1

)> 2n√

2nand a1 < k/w ≤ a2, k, w ≥ 2. Then,

CN(2, k, w : w(n))

UB0(2, k, w : w(n))≤ cn

− ln 2k(k−1)ln(1−w(w−1)

4k(k−1))

< ln

(1− 1

4

(wk

)2) cn

− ln(a21w2)

≤ ln

(1− 1

4a21

)cn

−2 ln a1 + 2n ln 2− ln 2n−→n→∞

−c ln(

1− 14a21

)2 ln 2

.

The fraction−c ln

(1− 1

4a21

)2 ln 2

� 1 for all pairs of values of c and a1 in Theorem 5.8.


UB_1

CB

SB

200 400 600 800 1000w

1 ´ 107

2 ´ 107

3 ´ 107

4 ´ 107

N

UB_1

CB

SB

200 400 600 800 1000w

1 ´ 107

2 ´ 107

3 ´ 107

4 ´ 107

N

UB1(2, 2w,w − 1: w), UB1(2, 1.81w,w − 1: w)CN(w) = 6(w − 1)2, CN(w) = 6(w − 1)2,SB(2, 2w,w − 1: w) SB(2, 1.81w,w − 1: w)

UB_1

CB

SB

200 400 600 800 1000w

1 ´ 1010

2 ´ 1010

3 ´ 1010

4 ´ 1010

N

UB_1

CN

SB

200 400 600 800 1000w

1 ´ 1010

2 ´ 1010

3 ´ 1010

4 ´ 1010

N

UB1(3, 8w/5, w − 1: w), UB1(3, 17.1w/11, w − 1: w)CN(w) = 8(w − 1)3, CN(w) = 8(w − 1)3,SB(3, 8w/5, w − 1: w) SB(3, 17.1w/11, w − 1: w)

Figure 5.3: Comparison of the size of CARLs constructed in Theorems 5.7 and 5.9,CN(w), the upper bound UB1 given in Theorem 4.4, and SB, the Schonheim lowerbound in Theorem 4.2.


UB_1

UB_0

CN

SB

15 20 25 30 35 40n

100

200

300

400

500

N

UB_1

UB_ 0

CN

SB

15 20 25 30 35 40n

200

400

600

800

N

UB1(2, 3w(n)/2, 2: w(n)), UB1(2, 2w(n), 2: w(n)),UB0(2, 3w(n)/2, 2: w(n)), UB0(2, 2w(n), 2: w(n)),CN(n) = 3n, SB(2, 3w(n)/2, 2: w(n)) CN(n) = 6n, SB(2, 2w(n), 2: w(n))

Figure 5.4: Comparison of the size CN(n) of CARLs constructed in Theorem 5.8, theupper bound UB1 from Theorem 4.4, and the lower bound SB given in Theorem 4.2,when w(n) =

(n−1bn/2c−1

).

Chapter 6

Group divisible covering designs

with block size four

6.1 Introduction

In this chapter we present constructions of CARLs with strength t = 2 and row limit

w = 4. As mentioned in Chapter 3, when t = 2, CARLs are equivalent to group divisible

covering designs, GDCDs. Concurrent with the tradition in the combinatorial design

theory, in this chapter we construct CARLs with row limit four in their equivalent form

of 4−GDCDs.

From now on, we use the design notation. A k −GDCD of type gu is equivalent to

a CARL(2, u, g : k). The objective is to construct a GDCD with the smallest possible

number of blocks. Recall Definition 3.4 of the cover number C(k, gu) which is the smallest

number of blocks in a k−GDCD of type gu, that is C(k, gu) = CARLN(2, u, g : k). The

Schonheim lower bound in Theorem 4.2 for t = 2 and λ = 1 implies a lower bound on

C(k, gu), stated in the following corollary.

Corollary 6.1. [28]

C(k, gu) ≥⌈gu

k

⌈g(u− 1)

k − 1

⌉⌉.

70

Chapter 6. Group divisible covering designs with block size four 71

When the block size is a constant, Theorem 4.7 states that the lower bound in

Corollary 6.1 is asymptotically optimal. Indeed, for constant values of g and k, Theorem 4.7

states that C(k, gu) =(u2)

(k2)g2(1 + o(1)) as a function of u. Under the same conditions, the

lower bound in Corollary 6.1 is bounded as follows:

(u2

)(k2

)g2 ≤ ⌈guk

⌈g(u− 1)

k − 1

⌉⌉≤(u2

)(k2

)g2(1 +k − 1

(u− 1)g+

k(k − 1)

u(u− 1)g2

)=

(u2

)(k2

)g2(1 + o(1)).

Optimal GDCDs with block size three are known. Theorem 3.1 shows that for any

positive integers g and u ≥ 3, there exists a 3 − GDCD of type gu which meets the

Schonheim lower bound without exceptions. In this chapter, we prove the following

theorem regarding 4−GDCDs, which is stronger than the result of Theorem 4.7.

Theorem 6.2. There exists a positive integer δ, such that for any positive integer g and

u ≥ 4,

C(4, gu) ≤⌈gu

4

⌈g(u− 1)

3

⌉⌉+ δ,

except possibly when (1) g = 17 and u ≡ 0 (mod 3), or (2) g ≥ 8, g ≡ 2, 5 (mod 6), and

u ≡ 23 (mod 24) or u ∈ {29, 35, 41}.

Moreover, we prove that many families of 4 − GDCDs meet the Schonheim lower

bound (see Theorem 6.59) and we also construct two families of 4 − GDCDs which

exceed the Schonheim lower bound by at most 1 or 2 blocks (cf. Theorem 6.60). Our

constructions are recursive: if the ingredient 4−GDCDs meet the Schonheim lower bound

so does the resulting family of 4 − GDCDs. We are missing a number of ingredients

to completely solve this problem (see Appendix A). However, since exceptions usually

occur for objects with small parameters and our missing ingredients have relatively big

parameters, we make the following conjecture.


Conjecture 6.3. C(4, gu) =⌈gu4

⌈g(u−1)

3

⌉⌉+ δ where

δ =

1, when (g, u) ∈ {(1, 7), (1, 9), (1, 10), (2, 4), (6, 4)},

2, when (g, u) ∈ {(1, 19), (3, 6)},

0, otherwise.

Most of the exceptions in Conjecture 6.3 are already known. When g = 1, GDCDs

are equivalent to coverings with block size four, which are known (cf. Theorem 2.17). The

famous result about nonexistence of two mutually orthogonal Latin squares of order 2

and 6 (cf. Theorem 2.11) implies two more exceptions in the conjecture. Finally, we show

in Lemma 6.15 that the case (g, u) = (3, 6) follows from the case (g, u) = (1, 19).

Our objective is to construct optimal 4−GDCDs. With the conjecture in mind, in

the following sections, we say that there exists an optimal 4−GDCD of type gu if there

exists a 4−GDCD of type gu having⌈gu4

⌈g(u−1)

3

⌉⌉blocks. Alternatively, we also state

that C(4, gu) =⌈gu4

⌈g(u−1)

3

⌉⌉where appropriate.

The foreknowledge of the number of blocks of a 4−GDCD which we want to obtain

gives us the structure of the excess graph which such a 4 − GDCD should have. In

Section 6.2 we show that if the size of a 4−GDCD meets the Schonheim lower bound,

then the 4−GDCD has an empty, or 1-regular, or 2-regular excess graph, or its excess

graph is within a few edges of one of these three cases (see Table 6.1). The structure of the

excess graph of the result of a construction indicates the permissible structure of the excess

graphs of the ingredients in the construction. We define essential and auxiliary ingredients

and recall Wilson’s construction for GDCDs in Section 6.2. In Sections 6.3-6.5, we reduce

the problem of finding an optimal 4−GDCD of type gu to proving the existence of an

optimal solution for a 4−GDCD of type gu for finitely many small values of g and u.

Furthermore, we construct an optimal solution to most of the families of 4 − GDCDs

which have a regular excess graph, as well as to some families of those whose excess graph

is not regular.


Section 6.4 considers constructions of families of 4−GDCDs with a fixed number of

columns u and variable group sizes g (mod 12). In Section 6.5, we provide constructions

for 4 − GDCDs with a constant g and variable u (mod 12). We refer a reader to the

summary of constructions in Table 6.2 in Section 6.8, which may help the reader to follow

the results easier. In Section 6.6, we present an alternative construction for a family of

4−GDCDs which requires fewer ingredients, but it is harder to find these ingredients

computationally. We finish this chapter with a summary of results and constructions.

6.2 Constructions and notation

In Section 5.3, we already defined and applied Wilson’s construction for CARLs. Originally,

Wilson’s construction is used for group divisible designs [54]. Here, we state the equivalent

of Construction 5.6 for group divisible covering designs. In this section, we also introduce

the notions of essential and auxiliary ingredients for the constructions.

Construction 6.4. Given positive integers g, u and k, such that u ≥ k, if there exists

(V,B), a PBD(u,K, λ1), and if for every k0 ∈ K there exists a (k, λ2)−GDCD of type

gk0 with Nk0 blocks, then there exists a (k, λ1λ2) − GDCD of type gu with∑

B∈BN|B|

blocks. (Note that for all B ∈ B, |B| = k0 ∈ K.)

Proof. Replace (inflate) each element of V by g elements, which are now going to be the

groups of the desired GDCD. For each block B ∈ B, put an isomorphic copy of the given

(k, λ2)−GDCD of type g|B| on the respective groups of elements.

We introduce the notions of essential and auxiliary ingredients by an example. Assume

we are given a positive integer g, g 6≡ 3, 5 (mod 6) and g 6∈ {2, 6}. To construct a family

of 4−GDCDs of type gu0 where u0 = 6u+ 9, u ≥ 4, we can apply Wilson’s construction

on a family of 4−GDDs of type 6u91, which exists by Theorem 2.8 for all u ≥ 4. Viewing

the GDD as a PBD(6u + 9, {4, 6, 9}, 1), it has only one block of size 9 and hence we


require only one copy of an ingredient 4−GDCD of type g9 for Construction 6.4. On

the other hand, the number of blocks of size 4 and 6 of the PBD depends on u0; we

require multiple copies of a 4−GDCD of type g4 and a 4−GDCD of type g6. If the

ingredient 4−GDCDs with 4 or 6 groups meet the Schonheim lower bound, then the

resulting GDCD with u0 groups exceeds the Schonheim lower bound by as many blocks

as the ingredient 4−GDCD with 9 groups does. Hence, the ingredient 4−GDCDs of

types g4 and g6 are essential since the number of times they are used in the construction

depends on u0. The ingredient 4−GDCD of type g9 is auxiliary since the number of its

copies required for the construction is independent of the total number of groups, u0.

In the above example, the number of groups u0 is a variable, and the group size g is

fixed. In Section 6.4, we present constructions in which the number of groups is fixed

and the group size is a variable. The concept of the essential and auxiliary ingredients is

similar in this case. We state it more formally below.

Let k be an integer. Given a construction of a family of k −GDCD of type gu where

x ∈ {g, u} is a variable parameter and the other parameter is a constant, we divide the

ingredient GDCDs for the construction in two kinds:

1. if the number of copies of the ingredient GDCD for the construction depends on x,

then the GDCD is an essential ingredient,

2. if the number of copies of the ingredient GDCD for the construction is independent

of x, then the GDCD is an auxiliary ingredient.

Constructions in Sections 6.4 and 6.5 have at most one auxiliary ingredient.

Table 6.1 gives the number of edges in the excess graph of a 4−GDCD of type gu

which have⌈gu4

⌈g(u−1)

3

⌉⌉blocks, assuming it exists. Indeed, given the number of blocks,

one can compute the degree sequence of the excess graph of the GDCD, which determines

the number of edges in the graph. If the excess graph in Table 6.1 has gu2

edges, then it is

a 1−regular graph; if the excess graph has gu edges, then it is a 2−regular graph. All


other cases are within a few edges of an empty graph, a 1− or a 2−regular graph.

u\g 0 1 2 3 4 5 6 7 8 9 10 11

0 0 gu2† gu‡ 0 gu

2† gu‡ 0 gu

2† gu‡ 0 gu

2† gu‡

1 0 0 0 0 0 0 0 0 0 0 0 0

2 0 gu+ 3 gu2† 3 gu‡ gu

2† 0 gu+ 3 gu

2† 3 gu‡ gu

2†

3 0 gu+32

gu‡ 3 gu2

gu‡ 0 gu+92

gu‡ 3 gu+62

gu‡

4 0 0 0 0 0 0 0 0 0 0 0 05 0 gu+ 3 gu+6

20 gu‡ gu+3

20 gu+ 3 gu

20 gu‡ gu+3

2

6 0 gu2† gu‡ 3 gu

2† gu+ 3 0 gu

2† gu‡ 3 gu

2† gu+ 3

7 0 3 0 3 0 3 0 3 0 3 0 3

8 0 gu‡ gu2† 0 gu‡ gu

2† 0 gu‡ gu

2† 0 gu‡ gu

2†

9 0 gu+32

gu‡ 0 gu2

gu+ 3 0 gu+92

gu‡ 0 gu+62

gu+ 310 0 3 0 3 0 3 0 3 0 3 0 311 0 gu‡ gu+6

23 gu‡ gu+3

20 gu‡ gu

23 gu‡ gu+3

2

Table 6.1: The number of edges in an excess graph of a 4 − GDCD of type gu having⌈gu4

⌈g(u−1)

3

⌉⌉blocks. The columns are congruence classes g (mod 12) and the rows are

the congruence classes u (mod 12). The excess graph is 1-regular if marked with a (†),or 2-regular if marked by a (‡).

The structure of the excess graph of the desired 4−GDCD which we are constructing

determines the essential and auxiliary objects which can be used. For example, a

4−GDCD of type gu where g ≡ 1 (mod 12) and u ≡ 5 (mod 12) would have an excess

graph in which two vertices have degree five and all other vertices have degree two.

Therefore, in a construction of such a 4−GDCD, the essential ingredients have to have

a 2-regular excess graph, as they appear multiple times, and there has to be one auxiliary

ingredient such that in its excess graph, two vertices have degree five and all other vertices

have degree two. For the same reason, the earlier presented construction of 4−GDCDs

of type gu0 where u ≡ 3 (mod 6) fails if g ≡ 3, 5 (mod 6). Indeed, a 4−GDCD of type

g6 is an essential ingredient, but it does not have a regular excess graph in these cases.

Since we want to construct 4−GDCDs which meet the Schonheim bound, we apply

Construction 6.4 to PBDs most of whose blocks have size four. Hence, we require an

essential ingredient 4−GDCD of type g4, which is actually a TD(4, g), and it exists by

Theorem 2.11 when g 6= 2, 6. Note that this family of objects has no edges in the excess


graphs. We consider the cases when g is equal to 2 or 6 separately.

6.3 g = 2 or 6

In this section we construct optimal 4−GDCD of types 6u and 2u, u ≥ 4, which meet

the Schonheim lower bound with two genuine exceptions already noted in Conjecture 6.3

and two possible exceptions given in Theorem 6.14.

A 4 − GDCD of type 6u has no edges in the excess graph. In other words, it is a

4−GDD of type 6u, which exists for all u > 4 by Theorem 2.4. When u = 4, there exists

a covering array CA(37; 2, 4, 6) [14, Table IV.10.22], which is equivalent to a 4−GDCD

of type 64 with 37 blocks. We can also construct this object by filling the hole of an ITD

of type (6, 2)4 with an optimal 4−GDCD of type 24, which is given below Theorem 6.5.

This is the closest one can get to the lower bound and not contradict Theorem 2.11.

Theorem 6.5. C(4, 6u) =⌈6u4

⌈6(u−1)

3

⌉⌉+ δ = 3u(u− 1) + δ where

δ =

1, when u = 4

0, otherwise.

Now, we construct a 4 − GDCD of type 2u. If u ≡ 1 (mod 3), u > 4, there

exists a 4 − GDD of type 2u by Theorem 2.4. The case when u = 4 is similar to

the case of 4 − GDCD of type 64 above. The lower bound cannot be met since

there do not exist two MOLS of order 2 (cf. Theorem 2.11), but one can easily

find a 4 − GDCD of type 24 with 5 blocks. For example, let the set of elements

be V = Z8, and let groups be G = {{0, 4}, {1, 5}, {2, 6}, {3, 7}}. Then, take B =

{{0, 1, 2, 3}, {0, 1, 2, 7}, {0, 3, 5, 6}, {1, 3, 4, 6}, {2, 4, 5, 7}}. Hence, we get the following

lemma.


Lemma 6.6. When u ≡ 1 (mod 3), u ≥ 4, C(4, 2u) = u(u−1)3

+ δ where

δ =

1, when u = 4

0, otherwise.

Lemma 6.7. C(4, 2u) = u2

3when u ∈ {6, 9, 12, 15}.

Proof. We give an explicit solution for a 4 − GDCD of type 26 with an excess graph

isomorphic to two disjoint 6-cycles. Let V = Z6 × Z2 be the set of elements. Partition

V into six groups: Gi = {(i, x) : x ∈ Z2}, where i ∈ Z6. Let B be the following set of

blocks:

{(0, 0), (1, 0), (3, 1), (5, 0)}, {(0, 0), (1, 1), (2, 0), (5, 1)},

{(0, 1), (1, 0), (2, 1), (3, 0)}, {(0, 1), (1, 1), (3, 1), (5, 1)},

{(0, 0), (2, 1), (3, 1), (4, 1)}, {(0, 1), (2, 0), (4, 1), (5, 0)},

{(0, 1), (2, 1), (4, 0), (5, 1)}, {(0, 0), (2, 0), (3, 0), (4, 0)},

{(1, 0), (2, 0), (3, 1), (4, 0)}, {(1, 1), (2, 1), (4, 1), (5, 0)},

{(1, 0), (3, 0), (4, 1), (5, 1)}, {(1, 1), (3, 0), (4, 0), (5, 0)}.

In the following, we give the starter blocks of a 4 − GDCD of type 2u where u ∈

{9, 12, 15}.

When u = 9, 15, let the set of elements be {0, 1}×Zu. Let the group Gi = {(0, i), (1, i)},

where i ∈ Zu. Develop the following blocks +1 (mod u) in the second coordinate.

4−GDCD of type 29 whose excess graph consists of nine isolated edges of multiplicity

two:

{(1, 0), (1, 2), (1, 5), (0, 3)}

{(0, 0), (0, 7), (1, 3), (1, 4)}

{(0, 0), (0, 5), (0, 8), (1, 6)}


4−GDCD of type 215 whose excess graph is a cycle of length 30:

{(1, 0), (1, 12), (1, 13), (0, 5)}

{(0, 0), (0, 13), (0, 10), (1, 14)}

{(0, 0), (0, 4), (1, 14), (1, 6)}

{(1, 0), (1, 6), (1, 10), (0, 12)}

{(0, 0), (0, 6), (0, 14), (1, 11)}

When u = 12, let the set of elements be {0, 1} × Z12. However, let the group

Gij = {(i, j), (i, j + 6)}, where i ∈ {0, 1} and j ∈ {0, 1, . . . , 5} ⊂ Z12. Develop the

following blocks +1 (mod 12) in the second coordinate.

4−GDCD of type 212 whose excess graph is two disjoint 12-cycles:

{(0, 0), (0, 1), (1, 6), (1, 2)}

{(1, 0), (1, 1), (1, 3), (0, 3)}

{(0, 0), (0, 10), (0, 3), (1, 10)}

{(0, 0), (0, 8), (1, 4), (1, 11)}

Lemma 6.8. C(4, 2u) = u2

3for all u ≡ 0 (mod 3), u ≥ 18.

Proof. If u ≡ 0 (mod 3), u ≥ 18, there exists (V,G,B), a 4−GDD of type 2u−6111, by

Theorem 2.6. Let P = {{G1i , G

2i , G

3i } : Gj

i ∈ G, |Gji | = 2, i ∈ {1, 2, . . . , u−6

3}, j ∈ {1, 2, 3}}

be a partition of the set of groups in G of size two into disjoint subsets of size three.

Label elements in the groups of size two by Gji = {aGj

i, bGj

i}, Gj

i ∈ G. Take a new

element x, x 6∈ V , and construct new blocks C = {{x, aG1i, aG2

i, aG3

i}, {x, bG1

i, bG2

i, bG2

i} :

i ∈ {1, 2, . . . , u−63}}. Finally, adjoin x to the group of size eleven. On these elements,

construct an optimal 4 − GDCD of type 26, which exists by Lemma 6.7. Take these

blocks of the GDCD together with blocks in B ∪ C to get an optimal 4−GDCD of type

2u whose excess graph consists of 2u−63

disjoint 3-cycles and two disjoint 6-cycles.

It remains to consider the case when u ≡ 2 (mod 3). First, we give an example of an

optimal 4−GDCD of type 2641 which is an ingredient in the constructions.


Lemma 6.9. There exists an optimal 4−GDCD of type 2641 with 19 blocks such that

in the excess graph, the vertices corresponding to the elements in the groups of size two

have degree one and the vertices corresponding to the elements of the group of size four

have degree zero.

Proof. Let V = Z12 ∪ G, where G = {∞i : i ∈ Z4}. Let the groups of size two be

Gi = {i, i + 6} ⊂ V , and the group of size four be G. Then take the following set of

blocks:

{0, 1, 10, ∞0}

{2, 5, 6, ∞0}

{3, 7, 11, ∞0}

{4, 8, 9, ∞0}

{2, 9, 11, ∞1}

{3, 6, 10, ∞1}

{1, 4, 5, ∞1}

{0, 7, 8, ∞1}

{1, 8, 11, ∞2}

{4, 6, 7, ∞2}

{0, 2, 3, ∞2}

{5, 9, 10, ∞2}

{2, 7, 10, ∞3}

{3, 5, 8, ∞3}

{1, 6, 9, ∞3}

{0, 4, 11, ∞3}

{6, 8, 10, 11}

{1, 2, 3, 4}

{0, 5, 7, 9}.

Lemma 6.10. If there exists an optimal, auxiliary 4 − GDCD of type 28, then there

exists an optimal 4−GDCD of type 2u, where u ≡ 2 (mod 6) and u ≥ 26.

Proof. Let u = 6l + 2, where l ≥ 4. There exists a 4−GDD of type 12l by Theorem 2.4,

call it X. Add four infinite elements, G = {∞1,∞2,∞3,∞4}. Take each but one group

of size 12 of X, partition it into subsets of size two, and put a copy of the 4−GDCD

of type 2641 given in Lemma 6.9 on its elements and G, which is the group of size four.

Finally, partition both the last group of size 12 of X and G into subsets of size two, but

put a copy of an optimal 4−GDCD of type 28 on these elements.


Lemma 6.11. If there exists an optimal, auxiliary 4 − GDCD of type 25, then there

exists an optimal 4−GDCD of type 2u for all u ≡ 5 (mod 6), u ≥ 29.

Proof. Let u = 6l + 5, where l ≥ 4. There exists X, a 4 − GDD of type 12l61 by

Theorem 2.8, whose groups we can partition into subsets of size two. Add a group

G = {∞1,∞2,∞3,∞4}. On the elements of each group of size 12 and G put a copy of

an optimal 4−GDCD of type 2641 given in Lemma 6.9, such that G is the group with

four elements. Finally, partition G into two sets with two elements, and put a copy of an

optimal 4−GDCD of type 25 on the elements of G and the group of size 6 from X.

Lemma 6.12. There exists an optimal 4−GDCD of type 2u when u ∈ {5, 8, 11, 14, 20}.

Proof. We give the starter blocks of a 4−GDCD of type 2u when u ∈ {8, 14, 20}.

When u = 8, 20, let the set of points be {0, 1, 2, 3} × Zu/2. Let the group Gij =

{{(i, j), (i, j + u4)}, where i ∈ {0, 1, 2, 3} and j ∈ Zu/4. Develop the blocks +1 (mod u

2)

in the second coordinate.

g = 2 and u = 8:

{(0, 0), (1, 0), (2, 3), (3, 0)}

{(0, 0), (2, 1), (2, 2), (3, 1)}

{(0, 0), (0, 3), (1, 2), (2, 0)}

{(1, 0), (1, 3), (2, 0), (3, 2)}

{(0, 0), (1, 1), (3, 2), (3, 3)}

g = 2 and u = 20:

{(0, 0), (1, 3), (2, 2), (3, 7)}

{(0, 0), (0, 2), (2, 5), (2, 3)}

{(0, 0), (0, 7), (1, 1), (2, 4)}

{(0, 0), (2, 0), (2, 6), (3, 3)}

{(0, 0), (1, 8), (2, 8), (3, 4)}

{(1, 0), (1, 3), (2, 4), (3, 8)}

{(0, 0), (0, 9), (1, 5), (1, 9)}

{(1, 0), (1, 1), (3, 8), (3, 1)}

{(0, 0), (0, 4), (3, 0), (3, 2)}

{(0, 0), (1, 7), (2, 9), (3, 9)}

{(0, 0), (1, 2), (3, 5), (3, 1)}

{(1, 0), (1, 2), (2, 7), (2, 8)}

{(2, 0), (2, 3), (3, 2), (3, 1)}


When u = 14, let the set of elements be {0, 1, 2, 3} × Zu/2. Let the group Gij =

{(2i, j), (2i + 1, j)}, where i ∈ {0, 1} and j ∈ Zu/2. Develop the following blocks +1

(mod u2) in the second coordinate.

g = 2 and u = 14 :

{(2, 0), (2, 4), (3, 6), (3, 1)}

{(0, 0), (1, 4), (2, 3), (3, 1)}

{(0, 0), (0, 1), (1, 2), (1, 3)}

{(0, 0), (1, 6), (2, 2), (3, 5)}

{(0, 0), (1, 5), (2, 6), (3, 3)}

{(0, 0), (0, 4), (2, 4), (2, 5)}

{(0, 0), (0, 2), (3, 6), (3, 2)}

{(1, 0), (1, 4), (2, 4), (2, 2)}

{(1, 0), (1, 2), (3, 2), (3, 3)}

Finally, we give an explicit solution for a 4−GDCD of type 25 and a 4−GDCD of type

211, whose excess graphs are not regular. In each case, let the set of elements be Zgu, and

the group Gi = {i, i+ g}, where i ∈ Zu and (g, u) ∈ {(2, 5), (2, 11)}. Take the following

sets of blocks.

g = 2 and u = 5:

{0, 1, 2, 3} {0, 1, 4, 7} {1, 5, 8, 9} {2, 4, 5, 8}

{0, 2, 6, 9} {3, 4, 5, 6} {3, 5, 7, 9} {0, 6, 7, 8}

g = 2 and u = 11:

{7, 13, 11, 3}

{16, 7, 1, 9}

{20, 3, 4, 6}

{14, 18, 9, 8}

{2, 14, 6, 7}

{5, 20, 18, 11}

{5, 3, 21, 18}

{10, 4, 11, 14}

{0, 1, 15, 3}

{2, 16, 21, 4}

{17, 7, 0, 21}

{12, 5, 11, 9}


{20, 16, 12, 15}

{12, 6, 18, 13}

{13, 5, 15, 14}

{18, 10, 16, 17}

{5, 19, 17, 4}

{4, 12, 8, 7}

{0, 16, 14, 13}

{4, 0, 18, 1}

{2, 8, 20, 10}

{8, 0, 3, 16}

{6, 11, 16, 19}

{15, 19, 2, 18}

{12, 0, 10, 6}

{0, 7, 20, 19}

{17, 14, 20, 1}

{13, 8, 21, 20}

{17, 4, 13, 9}

{3, 2, 17, 12}

{5, 10, 15, 7}

{12, 19, 14, 21}

{9, 21, 6, 15}

{21, 1, 2, 11}

{2, 0, 9, 5}

{11, 15, 17, 8}

{9, 10, 3, 19}

{6, 1, 5, 8}

{19, 1, 10, 13}

Lemmas 6.10-6.12 imply the following corollary.

Corollary 6.13. There exists an optimal 4 − GDCD of type 2u for all u ≥ 4, u ≡ 2

(mod 3), except possibly when u ∈ {17, 23}.

We summarize the results of Lemmas 6.6 - 6.8 and Corollary 6.13 in the following

theorem.

Theorem 6.14. C(4, 2u) =⌈u2

⌈2(u−1)

3

⌉⌉+ δ for all u ≥ 4, where

δ =

1, when u = 4,

0, otherwise,

except possibly when u ∈ {17, 23}.


6.4 Constructions of 4−GDCD with a small number

of groups

In this section we adapt some of the constructions for the group divisible designs to

construct the GDCDs. Later, in Section 6.5, we apply Construction 6.4 using some of

the 4−GDCDs that we construct here as ingredients.

First, we consider the case when u ≡ 7, 10 (mod 12).

Lemma 6.15. There exists an optimal 4 − GDCD of type 3u when u ≡ 2, 3 (mod 4),

u > 6 with⌈3u(u−1)

4

⌉blocks. Moreover, an optimal 4−GDCD of type 36 has 25 blocks.

Proof. By Theorem 2.17, there exists an optimal 2− (3u+ 1, 4, 1) covering design when

3u+1 > 19, whose excess graph has only one edge of multiplicity three. Delete an element

whose corresponding vertex does not have any incident edges in the excess graph; blocks

which contained this element become u groups of size three. Take the remaining blocks

to be the blocks of the desired GDCD.

First we prove that n = C(4, 36) ≥ 25. Given an optimal 4−GDCD of type 36 with

n blocks, add an infinite point ∞ to the set of elements, adjoin ∞ to the groups of size

3 and take these groups as blocks to get a 2− (19, 4, 1) covering with n+ 6 blocks. By

Theorem 2.17, an optimal 2− (19, 4, 1) covering has 31 blocks and it must have an element

which belongs to exactly 6 blocks, call it x. Hence, n ≥ 25, and deleting x in the covering

yields a 4−GDCD of type 36 with 25 blocks.

Lemma 6.16. Let u ≡ 7, 10 (mod 12).

1. If g ≡ 0 (mod 2), then there exists an optimal 4−GDCD of type gu.

2. If g ≡ 1 (mod 2), g 6∈ {5, 7}, and there exists an optimal, auxiliary 4−GDCD of

type 3u, then there exists an optimal 4−GDCD of type gu.


Proof. When g ≡ 0 (mod 2), there exists a 4−GDD of type gu by Theorem 2.4. The case

g = 1 is given in Theorem 2.17. When g ≡ 1 (mod 2), g ≥ 9, there exists a 4− IGDD

of type (g, 3)u by Theorem 2.13. Fill the hole with an optimal 4−GDCD of type 3u.

Together Lemma 6.15 and Lemma 6.16 imply the following theorem.

Theorem 6.17. Let u ≡ 7, 10 (mod 12) and let g be any positive integer, g 6= 5, 7. There

exists an optimal 4−GDCD of type gu of size⌈gu4

⌈g(u−1)

3

⌉⌉+ δ, where

δ =

2, when (g, u) = (1, 19)

1, when (g, u) ∈ {(1, 7), (1, 10)}

0, otherwise.

We give a construction of a 4 − GDCD of type gu when g = 5, 7 and u ≡ 7, 10

(mod 12) in Corollary 6.43.

In the following, for u ≥ 4, we consider cases with respect to congruence classes of g

(mod 6).

Theorem 2.4 directly implies the following lemma:

Lemma 6.18. There exists an optimal 4−GDD of type gu when u ≥ 4 and

1. g ≡ 0 (mod 6) except when (g, u) = (6, 4), or

2. g ≡ 3 (mod 6) and u ≡ 0, 1 (mod 4), or

3. any positive integer g and u ≡ 1, 4 (mod 12), except when (g, u) ∈ {(2, 4), (6, 4)}.

Lemma 6.19. Let u ≡ 2, 3 (mod 4). If there exists an optimal, auxiliary 4−GDCD of

type 3u, then there exists an optimal 4−GDCD of type gu for all g ≡ 3 (mod 6), except

possibly when (g, u) ∈ {(15, 14), (21, 14), (15, 18), (21, 18)}.

Proof. By Theorem 2.13, there exists a 4− IGDD of type (g, 3)u. Fill the hole with an

optimal 4−GDCD of type 3u constructed in Lemma 6.15.


Lemmas 6.15, 6.18 and 6.19 yield the following two theorems.

Theorem 6.20. Let g ≡ 3 (mod 6). C(4, gu) =⌈gu4

⌈g(u−1)

3

⌉⌉when

1. u ≥ 4 and u ≡ 0, 1 (mod 4),

2. u ≥ 7 and u ≡ 2, 3 (mod 4), except possibly when (g, u) ∈ {(15, 14), (21, 14),

(15, 18), (21, 18)}.

By Lemma 6.15, an optimal 4−GDCD of type 36 exceeds the Schonheim lower bound

by two blocks and applying the construction from Lemma 6.19 on this GDCD we get

a 4−GDCD of type g6 for any g ≡ 3 (mod 6) which exceeds the lower bound by two

blocks. Hence, we get the following theorem.

Theorem 6.21. Let g ≡ 3 (mod 6). Then, C(4, 36) = 25 and when g ≥ 9,

C(4, g6) ≤⌈

3g

2

⌈5g

3

⌉⌉+ 2.

6.4.1 Constructions using Double Group Divisible Designs

For further constructions, we need the existence of some double group divisible designs

which we obtain by Construction 2.16.

Now we consider three constructions for the case when g ≡ 1, 4 (mod 6).

Lemma 6.22. Let g ≥ 4, g ≡ 1, 4 (mod 6) and u ≡ 0, 6, 8, 11 (mod 12). Then

C(4, gu) =

⌈gu

4

⌈g(u− 1)

3

⌉⌉.

Proof. Since g−1 ≡ 0 (mod 3), by Theorem 2.15, there exists a 4−HGDD of type (u, 1g).

Fill each hole with an optimal 2− (u, 4, 1) covering, which exists by Theorem 2.17.

The following two lemmas regard the case g ≡ 1 (mod 3) and u ≡ 2, 3, 5, 9 (mod 12).


Lemma 6.23. Let u ≡ 2, 3, 5, 9 (mod 12), u ≥ 5. If there exists an optimal, auxiliary

4−GDCD of type mu for m ∈ {1, 4, 7, 10}, then there exists an optimal 4−GDCD of

type gu for g = 16, or g ≡ m (mod 12) and g ≥ 25.

Proof. Let g = 4(3l) + m where (l,m) = (1, 4) or l ≥ 2 and m ∈ {1, 4, 7, 10}. Apply

Construction 2.16 onto a 4−GDD of type 43lm1 with a 4−HGDD of type (u, 14), both

of which exist by Theorem 2.7 and Theorem 2.15, respectfully. We get a 4 − DGDD

of type (u, 43lm1). Fill the holes with an optimal essential TD(4, u) and an auxiliary

4−GDCD of type mu.

The following lemma further simplifies the problem for u ≡ 2, 5 (mod 12), u ≥ 17.

Lemma 6.24. Let g ≡ 1 (mod 3) and u ≡ 2, 5 (mod 12), u ≥ 17. If there exists an

optimal, auxiliary 4−GDCD of type g5, then there exists an optimal 4−GDCD of type

gu.

Proof. First, we construct an optimal 4 − GDCD of type 1(u−5)51. Since u − 1 ≡ 1, 4

(mod 12), by Theorem 2.1 there exists (V,B), a BIBD(u− 1, 4, 1). Let B ∈ B be a block,

and let P be a partition of V \B into subsets of size three. Add one infinite point, ∞, to

the set B to build a group of size five. Then, (B \ B) ∪ {P ∪ {∞} : P ∈ P} is a set of

blocks of the desired GDCD whose excess graph is a union of u−53

disjoint 3-cycles.

Now, if u ≡ 5 (mod 12), there exists a PBD(u, {4, 5, 5∗}, 1) [3]. On the other hand,

if u ≡ 2 (mod 12), there exists a PBD(u, {4, 5, 5∗, 6}, 1) [35]. In both cases, consider the

respective PBD as a GDD of type 1(u−5)51 and apply Construction 2.16 using it and

the 4−HGDD of type (g, 1b) where b ∈ {4, 5, 6}, which exists by Theorem 2.15. We get

a 4−DGDD of type (g, 1(u−5)51). Put a copy of the above constructed 4−GDCD of

type 1(u−5)51 on the elements of each group, take the holes to be new groups to get a

4−GDCD of type g(u−5)(5g)1. Finally, put a copy of the 4−GDCD of type g5 on the

elements of the group of size 5g.

Lemma 6.25. There exists an optimal 4−GDCD of type gu when:


1. g = 4 and u ∈ {9, 15}, or

2. g ∈ {4, 10, 22} and u ∈ {5, 14}.

Proof. We develop these 4−GDCDs from starter blocks.

When g = 4 and u ∈ {5, 9, 15}, let the set of elements be Z4 × Zu. Let the group

Gi = Z4 × {i}, where i ∈ Zu. Develop the following starter blocks +1 (mod u) in the

second coordinate.

g = 4 and u = 5:

{(1, 0), (1, 3), (2, 2), (2, 1)}

{(0, 0), (0, 1), (2, 2), (2, 4)}

{(1, 0), (1, 4), (3, 1), (3, 3)}

{(0, 0), (0, 1), (3, 2), (3, 4)}

{(0, 0), (0, 3), (1, 2), (1, 1)}

{(2, 0), (2, 3), (3, 1), (3, 2)}

g = 4 and u = 9:

{(0, 0), (0, 7), (1, 6), (2, 4)}

{(0, 0), (1, 5), (2, 4), (3, 8)}

{(1, 0), (1, 5), (3, 2), (3, 3)}

{(0, 0), (2, 2), (2, 7), (3, 1)}

{(0, 0), (0, 8), (1, 1), (1, 3)}

{(2, 0), (2, 8), (3, 1), (3, 5)}

{(0, 0), (0, 3), (3, 7), (3, 5)}

{(0, 0), (0, 5), (2, 1), (2, 8)}

{(1, 0), (1, 8), (2, 4), (2, 1)}

{(0, 0), (1, 7), (3, 6), (3, 3)}

{(1, 0), (1, 3), (2, 6), (3, 4)}

g = 4 and u = 15:

{(0, 0), (1, 3), (2, 11), (3, 2)}

{(0, 0), (3, 1), (0, 11), (3, 9)}

{(0, 0), (1, 14), (0, 1), (2, 3)}

{(1, 0), (2, 5), (1, 2), (2, 13)}

{(0, 0), (3, 11), (0, 8), (3, 7)}

{(0, 0), (3, 6), (1, 12), (3, 8)}


{(0, 0), (1, 9), (2, 10), (3, 4)}

{(2, 0), (3, 11), (2, 9), (3, 1)}

{(0, 0), (2, 8), (0, 3), (2, 12)}

{(0, 0), (2, 1), (0, 10), (2, 14)}

{(0, 0), (1, 11), (0, 6), (1, 1)}

{(1, 0), (3, 12), (1, 9), (3, 13)}

{(1, 0), (2, 14), (1, 7), (2, 2)}

{(0, 0), (2, 7), (3, 10), (2, 6)}

{(0, 0), (1, 8), (0, 2), (1, 4)}

{(2, 0), (3, 5), (2, 10), (3, 8)}

{(0, 0), (1, 7), (2, 13), (3, 12)}

{(1, 0), (3, 8), (1, 1), (3, 2)}

{(1, 0), (2, 12), (1, 3), (3, 9)}

When g ∈ {10, 22} and u = 5, or g = 4 and u = 14, let the set of elements be Z2 × Zgu/2.

Let the group Gi = {(j, x) : j ∈ Z2, x ∈ Zgu/2, x ≡ i (mod u)}, where i ∈ Zu. Develop

the starter blocks +1 (mod gu/2) in the second coordinate.

g = 10 and u = 5:

{(0, 0), (0, 7), (0, 23), (1, 6)}

{(0, 0), (1, 8), (1, 2), (1, 9)}

{(0, 0), (0, 19), (0, 11), (1, 22)}

{(0, 0), (1, 19), (1, 11), (1, 7)}

{(0, 0), (0, 4), (0, 1), (1, 18)}

{(0, 0), (1, 12), (1, 21), (1, 23)}

{(0, 0), (0, 13), (1, 4), (1, 1)}

g = 22 and u = 5:

{(0, 0), (0, 54), (1, 12), (0, 46)}

{(0, 0), (1, 22), (0, 49), (1, 18)}

{(0, 0), (0, 38), (1, 46), (0, 14)}

{(0, 0), (0, 52), (1, 1), (0, 39)}

{(1, 0), (1, 46), (0, 13), (1, 39)}

{(0, 0), (0, 2), (1, 36), (0, 29)}

{(0, 0), (0, 36), (1, 29), (0, 43)}

{(1, 0), (1, 22), (0, 11), (1, 49)}

{(1, 0), (1, 3), (0, 39), (1, 41)}

{(1, 0), (1, 12), (0, 16), (1, 13)}

{(0, 0), (0, 33), (1, 39), (0, 51)}

{(0, 0), (0, 21), (1, 3), (0, 44)}


{(1, 0), (1, 34), (0, 42), (1, 36)}

{(1, 0), (1, 24), (0, 1), (1, 32)}

{(1, 0), (1, 44), (0, 46), (1, 18)}

g = 4 and u = 14:

{(0, 0), (1, 11), (1, 22), (1, 3)}

{(0, 0), (0, 22), (0, 4), (1, 2)}

{(0, 0), (1, 2), (1, 26), (1, 5)}

{(0, 0), (1, 15), (1, 10), (1, 16)}

{(0, 0), (0, 3), (0, 16), (1, 12)}

{(0, 0), (0, 26), (0, 5), (1, 23)}

{(0, 0), (0, 1), (1, 20), (1, 7)}

{(0, 0), (0, 19), (0, 11), (1, 4)}

{(0, 0), (1, 17), (1, 1), (1, 27)}

Finally, when g ∈ {10, 22} and u = 14, let the set of elements be Zgu. Let the group

Gi = {x ∈ Zgu : x ≡ i (mod u)}, where i ∈ Zu. Develop the following blocks +1

(mod gu).

g = 10 and u = 14:

{0, 59, 25, 93}

{0, 131, 21, 1}

{0, 96, 8, 2}

{0, 41, 24, 89}

{0, 83, 5, 105}

{0, 108, 18, 31}

{0, 101, 43, 104}

{0, 55, 71, 26}

{0, 15, 53, 64}

{0, 107, 4, 27}

{0, 19, 86, 12}

g = 22 and u = 14:

{0, 208, 225, 121}

{0, 85, 275, 241}

{0, 3, 55, 48}

{0, 276, 6, 8}

{0, 62, 203, 234}

{0, 13, 23, 39}

{0, 134, 194, 213}

{0, 242, 43, 227}

{0, 146, 82, 197}

{0, 183, 89, 147}

{0, 117, 220, 25}

{0, 20, 232, 293}


{0, 198, 133, 27}

{0, 127, 37, 180}

{0, 12, 120, 290}

{0, 24, 71, 235}

{0, 304, 173, 151}

{0, 192, 101, 267}

{0, 299, 245, 123}

{0, 68, 148, 307}

{0, 86, 249, 130}

{0, 262, 251, 93}

{0, 50, 99, 21}

{0, 5, 206, 77}

We summarize the above lemmas in the following two theorems.

Theorem 6.26. C(4, gu) =⌈gu4

⌈g(u−1)

3

⌉⌉when:

1. g ≡ 1, 4 (mod 6) and u ≡ 0, 6, 8, 11 (mod 12),

2. g ≡ 4 (mod 12) and u = 9, 15,

3. g ≡ 4 (mod 6) and u ≡ 2, 5 (mod 12),

4. g ≡ 1 (mod 12) and u ≡ 2, 3, 5, 9 (mod 12), except when (g, u) = (1, 9) and possibly

when g = 13, or u = 9 and g ≥ 25.

Also, when g ≡ 1 (mod 12), g 6= 13, there exists a 4−GDCD of type g9 with⌈9g4

⌈8g3

⌉⌉+1

blocks.

Proof. The first statement comes from Lemma 6.22. For the second statement, apply

Lemma 6.23 to an optimal 4 − GDCD obtained in the first statement of Lemma 6.25.

To get the third statement, first consider the case u = 5, 14. Applying Lemma 6.23 to an

optimal 4 − GDCD in the second part of Lemma 6.25, we get an optimal 4 − GDCD

of type gu for u = 5 or 14, and g ≡ 4, 10 (mod 12). Then, for any g ≡ 4 (mod 6), we

apply Lemma 6.24 to an optimal 4−GDCD of type g5 to get an optimal 4−GDCD of

type gu when u ≡ 2, 5 (mod 12), u ≥ 17. Also, since there exists an optimal 2− (u, 4, 1)

covering design by Theorem 2.17, Lemma 6.23 implies the fourth statement. Similarly, to


get the last statement, we apply Lemma 6.23 to an optimal 4−GDCD of type 19, given

in Theorem 2.17.

Theorem 6.27. Let g0 ∈ {7, 13, 19} and u0 ∈ {5, 14}. If there exists an optimal, auxiliary

4−GDCD of type g0u0 for all g0 and u0, then there exists an optimal 4−GDCD of type

gu where g ≡ 7 (mod 12) or g = 13, and u ≡ 2, 5 (mod 12).

Proof. If there exists an optimal 4−GDCD of type 75, by Lemma 6.23 there exists an

optimal 4−GDCD of type g5 where g ≡ 7 (mod 12), g 6= 19. To complete the spectrum,

we need an optimal 4−GDCD of types 195 and 135. Then, by Lemma 6.24, there exists

an optimal 4−GDCD of type gu for all g ≡ 7 (mod 12) or g = 13, and u ≥ 17, u ≡ 2, 5

(mod 12). The remaining cases are 4−GDCDs with u = 14. Again, the existence of an

optimal 4−GDCD of type 714 implies the existence of an optimal 4−GDCD of type

g14 for all g ≡ 7 (mod 12), g 6= 19, by Lemma 6.23.

We need 4 − GDCDs of type g6 and g9 as an essential and an auxiliary object,

respectively, to get any 4−GDCD of type gu where g ≡ 1 (mod 3), and u ≡ 3 (mod 6),

u ≥ 33 (cf. Lemma 6.44 and Theorem 6.46). Here, we state a corollary of Lemma 6.23

regarding objects in this class with u ≤ 27.

Corollary 6.28. Let u ∈ {9, 15, 21, 27}. If there exists an optimal, auxiliary 4−GDCD

of type gu0 for all g0 ∈ {4, 7, 10, 13, 19, 22}, then there exists an optimal 4 − GDCD of

type gu for all g ≡ 4, 7, 10 (mod 12) or g = 13.

Next we consider the case g ≡ 2 (mod 6).

Lemma 6.29. Let g ≡ 2 (mod 6). There exists an optimal 4 − GDCD of type gu for

all u ≥ 4, u 6≡ 5 (mod 6).

Proof. Theorem 6.14 gives a solution for the case g = 2.


Let g = 8. By Theorem 2.15, there exists a 4−HGDD of type (4, 2u). Put a copy of an

optimal 4−GDCD of type 2u on the elements of each group of the HGDD. Considering

the holes as new groups, we get an optimal 4−GDCD of type 8u.

When g ≥ 14, let g = 2(3l+ 1) for l ≥ 2. There exists a 4−HGDD of type (u, 2(3l+1))

by Theorem 2.15. Fill the holes with an optimal 4−GDCD of type 2u.

Lemma 6.30. Let u ≡ 5 (mod 6). If there exists an optimal, essential 4 − GDCD of

type 8u and an optimal, auxiliary 4−GDCD of type mu for m ∈ {2, 8, 14, 20}, then there

exists an optimal 4−GDCD of type gu for g = 32, or g ≡ m (mod 24), and g ≥ 50.

Proof. Let g = 8 · 3l + m where (l,m) = (1, 8), or l ≥ 2 and m ∈ {2, 8, 14, 20}. Since

there exist a 4−GDD of type 83lm1 by Theorem 2.9 and a 4−HGDD of type (u, 14),

Construction 2.16 yields a 4−DGDD of type (u, 83lm1). Fill the holes with an optimal,

essential 4−GDCD of type 8u, and an optimal, auxiliary 4−GDCD of type mu.

Lemma 6.31. There exists an optimal 4−GDCD of type gu when (g, u) ∈ {(8, 5), (20, 5),

(8, 11), (8, 17)}.

Proof. Let the set of elements be Zgu. Let the group Gi = {x ∈ Zgu : x ≡ i (mod u)},

where i ∈ Zu. Develop the following blocks +4 (mod gu).

g = 8 and u = 5:

{1, 39, 35, 27}

{1, 28, 4, 2}

{0, 4, 31, 32}

{3, 25, 24, 6}

{1, 20, 23, 9}

{0, 33, 26, 7}

{3, 14, 20, 37}

{3, 30, 1, 32}

{3, 34, 27, 26}

{2, 38, 36, 14}

{1, 17, 5, 14}


g = 20 and u = 5:

{2, 71, 33, 50}

{1, 74, 88, 50}

{1, 4, 8, 40}

{1, 62, 65, 54}

{1, 43, 60, 67}

{3, 62, 64, 81}

{1, 3, 95, 97}

{3, 29, 15, 57}

{1, 30, 89, 52}

{3, 14, 31, 32}

{3, 49, 16, 42}

{3, 25, 7, 39}

{3, 85, 69, 6}

{2, 60, 3, 44}

{2, 25, 51, 38}

{2, 30, 18, 64}

{0, 18, 74, 11}

{0, 29, 52, 21}

{3, 47, 56, 80}

{1, 2, 45, 69}

{3, 51, 70, 74}

{2, 34, 88, 0}

{2, 55, 93, 4}

{1, 20, 28, 47}

{2, 8, 11, 80}

{0, 53, 1, 44}

{2, 75, 59, 96}

g = 8 and u = 11

{0, 73, 15, 83}

{2, 29, 81, 16}

{2, 40, 69, 37}

{1, 40, 44, 19}

{1, 75, 82, 13}

{0, 72, 2, 10}

{0, 58, 78, 74}

{0, 59, 47, 86}

{2, 84, 50, 3}

{3, 53, 13, 48}

{1, 51, 3, 52}

{1, 85, 55, 21}

{1, 27, 50, 73}

{1, 30, 48, 6}

{3, 27, 86, 45}

{2, 15, 75, 58}

{0, 14, 42, 51}

{0, 35, 48, 21}

{1, 32, 0, 20}

{0, 9, 34, 17}

{0, 25, 23, 31}

{1, 2, 38, 8}

{0, 52, 71, 80}

{1, 83, 11, 14}

{1, 74, 29, 86}

{3, 0, 64, 38}

{3, 55, 59, 34}


g = 8 and u = 17:

{1, 41, 122, 98}

{0, 116, 104, 45}

{2, 20, 100, 133}

{0, 91, 47, 35}

{2, 14, 11, 75}

{0, 16, 9, 57}

{0, 13, 128, 95}

{3, 26, 73, 84}

{1, 125, 33, 62}

{1, 38, 5, 118}

{1, 26, 37, 17}

{2, 103, 85, 116}

{0, 90, 122, 94}

{3, 72, 111, 53}

{3, 12, 27, 76}

{1, 9, 11, 102}

{3, 120, 11, 85}

{0, 130, 114, 84}

{3, 133, 103, 109}

{1, 4, 15, 2}

{3, 21, 119, 96}

{3, 89, 79, 110}

{2, 74, 27, 79}

{0, 54, 74, 107}

{2, 69, 43, 68}

{0, 89, 131, 135}

{0, 2, 73, 59}

{1, 87, 61, 135}

{2, 99, 96, 9}

{1, 46, 28, 56}

{2, 80, 10, 86}

{3, 123, 30, 74}

{1, 39, 79, 100}

{3, 18, 45, 58}

{0, 25, 97, 92}

{2, 50, 71, 39}

{2, 32, 61, 131}

{3, 22, 117, 132}

{2, 56, 3, 60}

{2, 33, 5, 89}

{1, 115, 44, 84}

{2, 124, 76, 38}

{2, 28, 128, 52}

Recall that Theorem 6.14 proves the existence of an optimal 4−GDCD of type 2u

for any u ≥ 4, u 6∈ {17, 23}. Hence Lemmas 6.29-6.31 yield the following theorem.


⌈g(u−1)

3

⌉⌉when one of the following holds:

1. g ≡ 2 (mod 6) and u ≥ 4, u 6≡ 5 (mod 6), except when (g, u) = (2, 4),

2. u = 5 and g ≡ 8 (mod 12), except possibly when g = 44,

3. u ∈ {11, 17} and g ≡ 8 (mod 24),

4. u ∈ {5, 11} and g ≡ 2 (mod 24), except maybe when g = 26.


When u ≡ 5 (mod 6), u ≥ 53 and g ≡ 2 (mod 6), we use Wilson’s construction with

ingredients constructed in Lemma 6.30: an essential object having 8 groups and auxiliary

objects having 5, 11, 17 and 23 groups (cf. Lemma 6.49.) When u ≤ 47, Lemma 6.30

constructs these objects.

Corollary 6.33. Let u ∈ {5, 11, 17, 23, 29, 35, 41, 47}. If there exist an optimal, essential

4 − GDCD of type 8u and an optimal, auxiliary 4 − GDCD of type gu0 where g0 ∈

{2, 8, 14, 20}, then there exists an optimal 4 − GDCD of type gu for g = 32 or g ≡ g0

(mod 24), g ≥ 50.

When u ≡ 5 (mod 6), a 4 − GDCD of type gu has a 1-regular excess graph only

when g ≡ 8 (mod 12). Hence, a 4 − GDCD of type 8u is an essential ingredient

in Lemma 6.30 and later in Lemma 6.40 which regards the case g, u ≡ 5 (mod 6).

Therefore, Corollaries 6.33 and 6.41 require an optimal, essential 4−GDCD of type 8u

for u ∈ {23, 29, 35, 41, 47}, since Lemma 6.31 gives a solution when u ≤ 17. Unfortunately,

these essential 4−GDCDs are too large to be constructed by our computer programs.

Finally, we consider the case g ≡ 5 (mod 6).

Lemma 6.34. Let u ≥ 4, u 6≡ 5 (mod 6). If there exists an optimal, auxiliary 4−GDCD

of type 5u, then there exists an optimal 4 − GDCD of type gu for all g ≥ 23, g ≡ 5

(mod 6).

Proof. By Theorem 2.6, there exists a 4 − GDD of type 2( g−52

)51. Also, there exists

a 4 − HGDD of type (u, 14). Apply Construction 2.16 to get a 4 − DGDD of type

(u, 2( g−52

)51). Fill the holes with the optimal 4 − GDCD of type 2u, which exists by

Theorem 6.14, and an optimal 4−GDCD of type 5u.

Lemma 6.35. Let g ∈ {5, 11, 17}. There exists an optimal 4−GDCD of type gu when u ∈

{8, 12, 14, 15, 20, 24, 26} except possibly when (g, u) ∈ {(17, 12), (5, 26), (17, 26), (11, 24),

(17, 24)}.


Proof. When g = 5 and u = 12, let the set of elements be Z4 × Z15. Let the group

Gij = {(i, x) : x ∈ Z15, x ≡ j (mod 3)}, where i ∈ Z4 and j ∈ Z3. Develop the following

starter blocks +1 (mod 15) in the second coordinate.

g = 5 and u = 12:

{(0, 0), (1, 9), (2, 10), (3, 4)}

{(2, 0), (3, 11), (2, 9), (3, 1)}

{(0, 0), (2, 8), (0, 3), (2, 12)}

{(0, 0), (2, 1), (0, 10), (2, 14)}

{(0, 0), (1, 11), (0, 6), (1, 1)}

{(1, 0), (3, 12), (1, 9), (3, 13)}

{(1, 0), (2, 14), (1, 7), (2, 2)}

{(0, 0), (1, 3), (2, 11), (3, 2)}

{(0, 0), (3, 1), (0, 11), (3, 9)}

{(0, 0), (1, 14), (0, 1), (2, 3)}

{(1, 0), (2, 5), (1, 2), (2, 13)}

{(0, 0), (3, 11), (0, 8), (3, 7)}

{(0, 0), (3, 6), (1, 12), (3, 8)}

{(0, 0), (2, 7), (3, 10), (2, 6)}

{(0, 0), (1, 8), (0, 2), (1, 4)}

{(2, 0), (3, 5), (2, 10), (3, 8)}

{(0, 0), (1, 7), (2, 13), (3, 12)}

{(1, 0), (3, 8), (1, 1), (3, 2)}

{(1, 0), (2, 12), (1, 3), (3, 9)}

When g = 11 and u = 12, let the set of elements be Zgu. Also, let the group Gi = {x ∈

Zgu : x ≡ i (mod u)}, where i ∈ Zu. Develop the following blocks +4 (mod gu).

g = 11 and u = 12:

{2, 39, 79, 1}

{3, 80, 64, 103}

{1, 69, 48, 128}

{2, 105, 35, 63}

{2, 126, 77, 120}

{1, 103, 124, 38}

{3, 94, 78, 4}

{2, 33, 41, 106}

{0, 114, 17, 27}

{1, 122, 41, 59}

{3, 116, 76, 14}

{0, 15, 64, 35}

{0, 119, 38, 76}

{3, 28, 86, 69}

{2, 12, 130, 99}

{1, 0, 86, 34}

{0, 112, 11, 25}

{1, 4, 14, 70}

{0, 49, 128, 123}

{2, 131, 55, 42}

{2, 25, 32, 9}

{3, 55, 26, 11}

{3, 71, 25, 130}

{2, 70, 125, 23}


{1, 64, 32, 114}

{0, 79, 37, 63}

{1, 101, 120, 83}

{1, 26, 28, 35}

{2, 73, 8, 89}

{3, 131, 81, 37}

{1, 23, 56, 21}

{3, 127, 2, 129}

{0, 106, 62, 8}

{3, 118, 30, 121}

{0, 117, 88, 91}

{2, 115, 0, 22}

{1, 100, 29, 118}

{3, 68, 110, 84}

{1, 5, 57, 127}

{1, 54, 22, 76}

{0, 93, 104, 47}

When (g, u) ∈ {(5, 8), (11, 8), (17, 8), (5, 14), (17, 14), (11, 20)}, let the set of elements

be Z2 × Zgu/2. Let the group Gij = {(i, x) : x ∈ Zgu/2, x ≡ j (mod u/2)}, where

i ∈ Z2 and j ∈ Zu/2. Develop the following starter blocks +1 (mod gu/2) in the second

coordinate.

g = 5 and u = 8:

{(1, 0), (1, 3), (1, 10), (0, 5)}

{(0, 0), (0, 9), (0, 2), (1, 10)}

{(0, 0), (0, 14), (0, 15), (1, 11)}

{(1, 0), (1, 14), (1, 19), (0, 0)}

{(0, 0), (0, 17), (1, 6), (1, 4)}

{(0, 0), (0, 10), (1, 3), (1, 12)}

g = 11 and u = 8:

{(0, 0), (0, 26), (1, 41), (1, 12)}

{(0, 0), (0, 13), (0, 7), (1, 24)}

{(1, 0), (1, 14), (1, 25), (0, 24)}

{(0, 0), (0, 15), (0, 5), (1, 0)}

{(1, 0), (1, 35), (1, 13), (0, 31)}

{(0, 0), (0, 30), (1, 38), (1, 40)}

{(0, 0), (0, 11), (0, 2), (1, 9)}

{(0, 0), (0, 19), (0, 22), (1, 25)}

{(0, 0), (0, 23), (1, 16), (1, 22)}

{(0, 0), (0, 43), (1, 27), (1, 32)}

{(0, 0), (0, 17), (1, 35), (1, 36)}

{(1, 0), (1, 17), (1, 7), (0, 30)}

{(1, 0), (1, 26), (1, 23), (0, 21)}


g = 17 and u = 8:

{(0, 0), (0, 58), (1, 27), (1, 36)}

{(1, 0), (1, 29), (1, 54), (0, 53)}

{(1, 0), (1, 21), (1, 11), (0, 17)}

{(1, 0), (1, 2), (1, 33), (0, 7)}

{(0, 0), (0, 14), (0, 33), (1, 67)}

{(0, 0), (0, 1), (0, 62), (1, 7)}

{(0, 0), (0, 38), (1, 17), (1, 22)}

{(0, 0), (0, 50), (1, 0), (1, 30)}

{(0, 0), (0, 45), (1, 56), (1, 57)}

{(0, 0), (0, 11), (0, 9), (1, 40)}

{(0, 0), (0, 25), (1, 2), (1, 21)}

{(0, 0), (0, 47), (0, 13), (1, 55)}

{(0, 0), (0, 51), (1, 3), (1, 49)}

{(1, 0), (1, 65), (1, 15), (0, 40)}

{(0, 0), (0, 37), (1, 23), (1, 10)}

{(0, 0), (0, 63), (0, 41), (1, 33)}

{(1, 0), (1, 17), (1, 62), (0, 3)}

{(0, 0), (0, 39), (0, 42), (1, 58)}

{(0, 0), (0, 15), (1, 50), (1, 24)}

{(1, 0), (1, 7), (1, 41), (0, 36)}

g = 5 and u = 14:

{(1, 0), (1, 4), (1, 13), (0, 12)}

{(0, 0), (0, 9), (0, 34), (1, 5)}

{(1, 0), (1, 11), (1, 34), (0, 14)}

{(0, 0), (0, 24), (0, 19), (1, 14)}

{(1, 0), (1, 30), (1, 3), (0, 31)}

{(0, 0), (0, 3), (1, 22), (1, 3)}

{(1, 0), (1, 33), (1, 18), (0, 9)}

{(0, 0), (0, 31), (1, 8), (1, 33)}

{(0, 0), (0, 20), (0, 33), (1, 13)}

{(0, 0), (0, 27), (1, 10), (1, 16)}

{(0, 0), (0, 18), (0, 12), (1, 29)}

g = 17 and u = 14:

{(0, 0), (0, 92), (1, 11), (1, 80)}

{(0, 0), (0, 16), (1, 79), (1, 8)}

{(0, 0), (0, 53), (0, 85), (1, 82)}

{(0, 0), (0, 60), (0, 69), (1, 46)}

{(0, 0), (0, 79), (0, 24), (1, 77)}

{(1, 0), (1, 103), (1, 44), (0, 45)}

{(0, 0), (0, 51), (0, 43), (1, 25)}

{(0, 0), (0, 97), (1, 40), (1, 76)}

{(0, 0), (0, 23), (1, 90), (1, 23)}

{(0, 0), (0, 86), (0, 19), (1, 91)}


{(1, 0), (1, 74), (1, 111), (0, 109)}

{(1, 0), (1, 80), (1, 79), (0, 85)}

{(1, 0), (1, 86), (1, 117), (0, 15)}

{(1, 0), (1, 90), (1, 115), (0, 59)}

{(0, 0), (0, 10), (1, 59), (1, 97)}

{(0, 0), (0, 4), (0, 82), (1, 48)}

{(0, 0), (0, 104), (1, 21), (1, 3)}

{(0, 0), (0, 54), (0, 29), (1, 43)}

{(0, 0), (0, 75), (0, 57), (1, 81)}

{(1, 0), (1, 65), (1, 106), (0, 74)}

{(0, 0), (0, 58), (1, 33), (1, 7)}

{(1, 0), (1, 20), (1, 107), (0, 19)}

{(0, 0), (0, 47), (0, 11), (1, 20)}

{(0, 0), (0, 114), (1, 12), (1, 70)}

{(1, 0), (1, 22), (1, 73), (0, 89)}

{(0, 0), (0, 113), (0, 74), (1, 41)}

{(0, 0), (0, 46), (0, 48), (1, 61)}

{(1, 0), (1, 64), (1, 30), (0, 84)}

{(1, 0), (1, 66), (1, 43), (0, 50)}

{(0, 0), (0, 26), (0, 107), (1, 83)}

{(1, 0), (1, 95), (1, 92), (0, 41)}

{(0, 0), (0, 106), (1, 42), (1, 37)}

{(1, 0), (1, 108), (1, 102), (0, 80)}

{(0, 0), (0, 20), (0, 3), (1, 109)}

{(0, 0), (0, 30), (0, 118), (1, 26)}

{(1, 0), (1, 9), (1, 109), (0, 55)}

{(1, 0), (1, 62), (1, 15), (0, 115)}

g = 11 and u = 20:

{(0, 0), (0, 88), (0, 93), (1, 41)}

{(0, 0), (0, 82), (0, 97), (1, 69)}

{(0, 0), (0, 32), (0, 67), (1, 81)}

{(0, 0), (0, 24), (0, 79), (1, 40)}

{(1, 0), (1, 81), (1, 49), (0, 59)}

{(0, 0), (0, 96), (0, 87), (1, 95)}

{(0, 0), (0, 85), (0, 2), (1, 104)}

{(0, 0), (0, 66), (1, 24), (1, 11)}

{(0, 0), (0, 74), (1, 1), (1, 20)}

{(0, 0), (0, 49), (0, 102), (1, 62)}

{(1, 0), (1, 103), (1, 107), (0, 24)}

{(0, 0), (0, 77), (0, 26), (1, 9)}

{(1, 0), (1, 14), (1, 36), (0, 34)}

{(0, 0), (0, 62), (0, 69), (1, 57)}

{(0, 0), (0, 11), (1, 78), (1, 61)}

{(1, 0), (1, 65), (1, 64), (0, 21)}

{(1, 0), (1, 66), (1, 39), (0, 85)}

{(1, 0), (1, 57), (1, 2), (0, 9)}


{(1, 0), (1, 28), (1, 62), (0, 78)}

{(0, 0), (0, 68), (0, 56), (1, 31)}

{(1, 0), (1, 6), (1, 21), (0, 104)}

{(0, 0), (0, 71), (1, 106), (1, 33)}

{(1, 0), (1, 42), (1, 85), (0, 56)}

{(1, 0), (1, 59), (1, 77), (0, 103)}

{(0, 0), (0, 91), (0, 107), (1, 15)}

{(0, 0), (0, 1), (0, 65), (1, 88)}

{(1, 0), (1, 63), (1, 58), (0, 93)}

{(0, 0), (0, 6), (1, 59), (1, 3)}

{(0, 0), (0, 52), (0, 73), (1, 99)}

{(1, 0), (1, 72), (1, 41), (0, 105)}

{(1, 0), (1, 84), (1, 75), (0, 45)}

{(0, 0), (0, 34), (0, 38), (1, 38)}

{(1, 0), (1, 11), (1, 98), (0, 100)}

{(1, 0), (1, 24), (1, 8), (0, 82)}

{(0, 0), (0, 18), (0, 47), (1, 92)}

Finally, when (g, u) ∈ {(5, 15), (11, 15), (17, 15), (11, 14), (5, 20), (17, 20), (5, 24), (11, 26)},

the set of elements is Zgu. Let the group Gi = {x ∈ Zgu : x ≡ i (mod u)}, where i ∈ Zu.

Develop the following starter blocks +1 (mod gu).

g = 5 and u = 15:

{0, 12, 11, 18}

{0, 62, 20, 39}

{0, 72, 70, 24}

{0, 14, 10, 35}

{0, 34, 8, 66}

{0, 38, 22, 66}

g = 11 and u = 15:

{0, 162, 92, 146}

{0, 23, 6, 49}

{0, 63, 141, 74}

{0, 133, 18, 89}

{0, 64, 144, 132}

{0, 27, 82, 36}

{0, 112, 99, 151}

{0, 155, 48, 47}

{0, 88, 51, 86}

{0, 29, 7, 69}

{0, 119, 161, 127}

{0, 65, 93, 124}

{0, 5, 109, 25}


g = 17 and u = 15:

{0, 38, 136, 43}

{0, 222, 31, 230}

{0, 53, 190, 207}

{0, 194, 103, 171}

{0, 204, 185, 34}

{0, 245, 219, 218}

{0, 241, 97, 58}

{0, 13, 92, 169}

{0, 44, 66, 161}

{0, 124, 83, 28}

{0, 80, 167, 209}

{0, 81, 29, 11}

{0, 206, 143, 20}

{0, 148, 21, 155}

{0, 50, 54, 231}

{0, 106, 208, 141}

{0, 122, 116, 40}

{0, 2, 110, 142}

{0, 166, 57, 182}

{0, 246, 3, 62}

g = 11 and u = 14:

{0, 94, 127, 114}

{0, 152, 74, 81}

{0, 101, 77, 139}

{0, 107, 137, 115}

{0, 69, 90, 25}

{0, 87, 136, 135}

{0, 82, 31, 148}

{0, 91, 95, 41}

{0, 86, 122, 11}

{0, 108, 12, 9}

{0, 23, 125, 120}

{0, 93, 138, 128}

g = 5 and u = 20:

{0, 94, 71, 8}

{0, 13, 78, 11}

{0, 25, 4, 95}

{0, 88, 49, 64}

{0, 83, 10, 42}

{0, 99, 52, 55}

{0, 28, 46, 62}

{0, 31, 81, 74}

g = 17 and u = 20:

{0, 324, 53, 251}

{0, 126, 229, 312}

{0, 76, 228, 105}

{0, 246, 249, 145}

{0, 42, 163, 325}

{0, 331, 107, 132}

{0, 332, 139, 64}

{0, 49, 14, 93}

{0, 66, 328, 269}

{0, 288, 157, 306}

{0, 85, 62, 218}

{0, 115, 232, 2}


{0, 210, 171, 167}

{0, 321, 299, 65}

{0, 50, 63, 304}

{0, 192, 155, 67}

{0, 270, 243, 134}

{0, 95, 187, 5}

{0, 77, 196, 135}

{0, 165, 164, 211}

{0, 51, 68, 30}

{0, 292, 238, 124}

{0, 128, 161, 10}

{0, 6, 259, 314}

{0, 11, 213, 295}

{0, 190, 159, 333}

{0, 146, 244, 170}

g = 5 and u = 24:

{2, 12, 117, 57}

{3, 19, 44, 54}

{0, 27, 112, 49}

{1, 74, 86, 114}

{1, 91, 82, 55}

{0, 17, 66, 51}

{0, 103, 23, 117}

{0, 89, 53, 63}

{2, 81, 43, 61}

{1, 34, 36, 39}

{0, 54, 9, 1}

{1, 23, 24, 84}

{1, 10, 117, 41}

{1, 116, 111, 43}

{1, 29, 47, 40}

{2, 99, 29, 84}

{2, 22, 72, 90}

{3, 30, 67, 73}

{1, 38, 60, 13}

{2, 55, 59, 53}

{0, 77, 74, 93}

{0, 58, 16, 29}

{2, 32, 25, 76}

{2, 62, 15, 23}

{3, 22, 80, 112}

{2, 67, 0, 36}

{2, 63, 66, 44}

{0, 92, 55, 6}

{3, 109, 45, 46}

{0, 106, 75, 4}

{3, 106, 31, 2}

{0, 91, 111, 25}

{0, 100, 87, 94}

{3, 91, 61, 15}

{1, 48, 69, 88}

{0, 64, 26, 12}

{2, 35, 10, 46}

{1, 89, 110, 106}

{3, 84, 5, 63}

g = 11 and u = 26:

{0, 96, 223, 205}

{0, 206, 54, 45}

{0, 36, 70, 203}

{0, 60, 1, 214}

{0, 91, 207, 271}

{0, 209, 229, 108}

{0, 183, 270, 69}

{0, 259, 211, 88}

{0, 245, 102, 251}


{0, 283, 11, 166}

{0, 168, 224, 140}

{0, 246, 204, 136}

{0, 192, 93, 200}

{0, 39, 7, 17}

{0, 255, 43, 13}

{0, 262, 267, 71}

{0, 173, 61, 219}

{0, 76, 257, 21}

{0, 248, 126, 151}

{0, 53, 274, 145}

{0, 175, 37, 33}

{0, 284, 186, 237}

{0, 23, 89, 147}

Lemmas 6.34 and 6.35 yield the following theorem. We use these objects in an

application of Wilson’s construction in Lemmas 6.45 and 6.49 to construct families of

4−GDCDs of type gu where g ≡ 5 (mod 6) and u ≡ 0 (mod 3), u ≥ 48, and g = 11, 17

and u ≡ 2 (mod 3). Here we also state a corollary of Lemma 6.34 for u ≤ 26.

Theorem 6.36. Let u ∈ {8, 12, 14, 15, 20, 24, 26}. There exists an optimal 4−GDCD of

type gu for all g ≡ 5 (mod 6), except possibly when (g, u) ∈ {(17, 12), (5, 26), (17, 26),

(11, 24), (17, 24)}.

Corollary 6.37. Let u ∈ {18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 51}. If there exists an

optimal, auxiliary 4−GDCD of type 5u, then there exists an optimal 4−GDCD of type

gu for all g ≡ 5 (mod 6), g 6= 11, 17.

Now we construct optimal 4−GDCDs of type gu where g ≡ 5 (mod 6), g 6= 11, 17,

and u ≡ 2 (mod 6) using a similar construction method as in Lemma 6.10.

Lemma 6.38. There exists an optimal 4−GDCD of type 56101 such that in its excess

graph, the vertices corresponding to the elements in the groups of size five have degree one,

and the vertices corresponding to the elements in the group of size ten have degree zero.

Proof. Let the set of elements be V = Z30 ∪ X, where X = {∞i : i ∈ {0, 1, . . . , 9}}.


Develop the following blocks +6 (mod 30).

{5, 6, 16, ∞0}

{2, 27, 19, ∞0}

{5, 4, 7, ∞1}

{0, 2, 3, ∞1}

{4, 15, 8, ∞2}

{5, 25, 0, ∞2}

{3, 14, 5, ∞3}

{0, 16, 1, ∞3}

{0, 8, 17, ∞4}

{3, 4, 1, ∞4}

{0, 19, 15, ∞5}

{5, 8, 10, ∞5}

{4, 21, 12, ∞6}

{5, 2, 13, ∞6}

{5, 15, 22, ∞7}

{1, 8, 18, ∞7}

{3, 6, 29, ∞8}

{2, 22, 1, ∞8}

{0, 28, 14, ∞9}

{3, 17, 13, ∞9}

{4, 18, 9, 25}

{0, 4, 11, 26}

{5, 14, 27, 19}

Theorem 6.39. Let g ≡ 5 (mod 6), g 6= 11, 17, and u ≡ 2 (mod 6), u ≥ 4. Then, there

exists an optimal 4−GDCD of type gu.

Proof. Theorem 6.36 gives a solution when u ∈ {8, 14, 20}. Let u ≥ 26, u = 6l + 2, l ≥ 4.

First, we construct an optimal 4−GDCD of type 5u. Take a copy of a 4−GDD of type

30l, which exists by Theorem 2.4, and add the set X = {∞i : i ∈ {0, 1, . . . , 9}} of ten

infinite elements. Partition each but one group of thirty elements into subsets of size five

and on them and the set X put the 4−GDCD of type 56101 from Lemma 6.38. Partition

the last group of thirty elements into subsets of size five, and also partition X into two

subsets of size five, and put on these elements a copy of an optimal 4−GDCD of type

58 which exists by Lemma 6.35. Hence, we get an optimal 4−GDCD of type 5u. Now,

we can apply Lemma 6.34 on the 4−GDCD of type 5u to get an optimal 4−GDCD of

type gu for all g ≥ 23.


Lemma 6.40. Let u ≡ 5 (mod 6). If there exist an optimal essential 4−GDCD of type

8u and an auxiliary 4−GDCD of type mu for every m ∈ {5, 11, 17, 23}, then there exists

an optimal 4−GDCD of type gu for all g ≥ 53, g ≡ m (mod 24).

Proof. Let g = 8(3l) +m for l ≥ 2 and m ∈ {5, 11, 17, 23}. Apply Construction 2.16 with

a 4−GDD of type 83lm1 and a 4−HGDD of type (u, 14), which exist by Theorem 2.9

and Theorem 2.15, respectively, to obtain a 4−DGDD of type (u, 83lm1). Fill the holes

with an optimal 4−GDCD of type 8u and an optimal 4−GDCD of type mu.

The GDCDs constructed by Lemma 6.40 are ingredients for an application of Wilson’s

constructions in Lemma 6.49. Also, we use it to construct optimal solution of objects

with smaller number of groups in the following corollary.

Corollary 6.41. Let u ∈ {5, 11, 17, 23, 29, 35, 41, 47}. If there exist an optimal, essential

4 − GDCD of type 8u and an optimal, auxiliary 4 − GDCD of type gu0 where g0 ∈

{5, 11, 17, 23}, then there exists an optimal 4−GDCD of type gu for all g ≡ g0 (mod 24),

g ≥ 53.

6.5 4−GDCD of type gu where g 6= 1, 2, or 6

To complete the study, we use some of the families of 4−GDCDs obtained in Section 6.4

as ingredients in Construction 6.4.

When g 6∈ {2, 6}, a 4−GDCD of type g4 is a TD(4, g), which exists by Theorem 2.11.

This family of 4− GDCDs is the family of essential ingredients in all of the following

applications of Construction 6.4. We assume this fact in all proofs involving Construc-

tion 6.4 in this section, and we do not repeat it further. Moreover, a TD(4, g) when

g 6∈ {2, 6} has an empty excess graph, so they do not contribute any edges to the excess

graph of the solutions.

In the following we consider the cases with respect to u (mod 12).


Lemma 6.42. Let g 6∈ {1, 2, 6}. If there exists an optimal, auxiliary 4−GDCD of type

g7, then there exists an optimal 4−GDCD of type gu for all u ≥ 22, u ≡ 7, 10 (mod 12).

Proof. When u ≥ 22, u ≡ 7, 10 (mod 12), there exists a PBD(u, {4, 7∗}, 1) by Theo-

rem 2.2. Apply Construction 6.4 on this PBD using a TD(4, g) as an essential ingredient,

and the optimal 4−GDCD of type g7 as an auxiliary ingredient.

Lemma 6.42 completes the missing cases of Theorem 6.17.

Corollary 6.43. Let u ≡ 7, 10 (mod 12), u 6∈ {10, 19} and let g0 ∈ {5, 7}. If there

exists an optimal, auxiliary 4−GDCD of type g07 for all g0, then there exists an optimal

4−GDCD of type gu0 .

Now we consider two cases when u ≡ 0 (mod 3).

Lemma 6.44. Let g 6≡ 3, 5 (mod 6), g 6∈ {1, 2, 6}. If there exist an optimal, essential

4−GDCD of type g6 and an optimal, auxiliary 4−GDCD of type gm where m ∈ {0, 9},

then there exists an optimal 4−GDCD of type gu for all u ≥ 30, u ≡ m (mod 6).

Proof. When u ≡ 0 (mod 6), there exists a 4−GDD of type 6u6 by Theorem 2.4. On the

other hand, when u ≡ 3 (mod 6), there exists a 4−GDD of type 6(u−96

)91 by Theorem 2.8.

Apply Construction 6.4 on these designs using the optimal 4−GDCD of type g6 as the

essential ingredient, and the optimal 4−GDCD of type g9 as the auxiliary GDCD.

Lemma 6.45. Let g ≡ 3, 5 (mod 6), g 6∈ {1, 2, 6}. If there exist an optimal, essential

4 − GDCD of type g12 and an optimal, auxiliary 4 − GDCD of type gm where m ∈

{0, 6, 9, 15}, then there exists an optimal 4− GDCD of type gu for all u ≥ 48, u 6= 51,

and u ≡ m (mod 12).

Proof. There exists a 4−GDD of type 12(u−m12 )m1 by Lemma 2.10. Use it as the PBD

in Construction 6.4 with the 4−GDCD of type g12 as the essential ingredient, and the

4−GDCD of type gm as the auxiliary GDCD.


We can apply Lemmas 6.44 and 6.45 using some of the results in Section 6.4 as

ingredients to construct in the worst case a close to optimal solution for 4 − GDCDs

of type gu when g ≡ 1 (mod 3), g ≥ 25, u ≡ 3 (mod 6), u ≥ 39, and g ≡ 5 (mod 6),

g ≥ 23, u ≡ 0 (mod 3), u ≥ 48.

Theorem 6.46. Let u ≡ 3 (mod 6), u ≥ 4.

1. When g ≡ 4 (mod 12) and u 6∈ {21, 27}, there exists an optimal 4−GDCD of type

gu.

2. If there exists an optimal, auxiliary 4−GDCD of type g09 for all g0 ∈ {7, 10}, then

there exists an optimal 4−GDCD of type gu where g 6∈ {19, 22}, g ≡ g0 (mod 12)

and u 6∈ {15, 21, 27}.

3. If there exists an optimal, auxiliary 4−GDCD of type g09 for all g0 ∈ {13, 19, 22},

then there exists an optimal 4 − GDCD of type gu0 , except possibly when u ∈

{15, 21, 27}.

Proof. The first statement follows directly from Theorem 6.26 and Lemma 6.44. By

Lemma 6.23, the existence of an optimal 4−GDCD of type g0u for g0 = 7 and g0 = 10

implies the existence of an optimal 4 − GDCD of type g9 for all g ≡ 7, 10 (mod 12),

g ≥ 31. Hence, the existence of an optimal 4−GDCD of type g19 for g1 ∈ {13, 19, 22}

completes the solution for u = 9, which is an ingredient for Lemma 6.44.

Recall that if g ≡ 1 (mod 3), g ≥ 25, and u ≡ 3 (mod 6), u ≤ 27, Lemma 6.23

provides a construction of a 4 − GDCD of type gu which depends on some auxiliary

ingredients (cf. Corollary 6.28).

Theorem 6.47. Let g ≡ 5 (mod 6).

1. There exists an optimal 4 − GDCD of type gu for all u ≡ 0, 3 (mod 12), except

possibly when g = 17 or u ∈ {27, 36, 39, 51} or (g, u) = (11, 24).


2. If there exists an optimal, essential 4 − GDCD of type 1712, then there exists

an optimal 4 − GDCD of type 17u for all u ≡ 0, 3 (mod 12), except possibly for

u ∈ {24, 27, 36, 39, 51}.

3. Let u0 ∈ {6, 9}. If there exists an auxiliary 4 − GDCD of type 5u0, then there

exists an optimal 4 − GDCD of type gu for all u ≡ u0 (mod 12), u ≥ 54, and

g 6∈ {11, 17}.

4. Let u0 ∈ {6, 9}. If there exist an optimal, essential 4−GDCD of type 1712 and an

optimal, auxiliary 4−GDCD of type g0u0 for all g0 ∈ {11, 17}, then there exists an

optimal 4−GDCD of type gu0 for all u ≡ u0 (mod 12), u ≥ 54.

Proof. Theorem 6.36 and Lemma 6.45 imply the first two statements. Indeed, Theo-

rem 6.36 gives a solution to an optimal 4−GDCD of type g24 when g 6∈ {11, 17}. The

4−GDCD of type 1712 is one of the essential ingredients which requires too many blocks

for our algorithms to find. If we were to have it, Lemma 6.45 would imply the second

claim. When u0 ∈ {6, 9}, given the existence of an optimal 4−GDCD of type g0u0 for

all g0 ∈ {5, 11, 17}, there exists an optimal 4−GDCD of type gu0 for all g ≡ 5 (mod 6)

by Lemma 6.34. Then Lemma 6.45 implies the third and forth statements.

Recall, Corollary 6.37 gives a construction of a 4 − GDCD of type gu when g ≡ 5

(mod 6), g ≥ 23, and u ∈ {18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 51}.

Now we consider the case u ≡ 2 (mod 3). First, Lemma 6.18 and Theorems 6.26,

6.32, and 6.36 give the existence of an optimal 4−GDCD when u = 8.

Lemma 6.48. There exists an optimal 4−GDCD of type g8 for all positive integers g.

Lemma 6.49. Let g be a positive integer. If there exists an optimal, auxiliary 4−GDCD

of type gm where m ∈ {5, 8, 11, 14, 17, 20, 23, 26}, then there exists an optimal 4−GDCD

of type gu when u = 32 or u ≥ 53, u ≡ m (mod 24).


Proof. If u ≡ 8 (mod 24), there exists a 4−GDD of type 8u8 by Theorem 2.4. Otherwise,

u ≡ m (mod 24), m ∈ {5, 11, 14, 17, 20, 23, 26}, and there exists a 4 − GDD of type

8(u−m8 )m1 by Theorem 2.9. Apply Construction 6.4 on the GDD, the essential 4−GDCD

of type g8 given in Lemma 6.48 and the auxiliary 4−GDCD of type gm.

When g ≡ 2 (mod 6) and u ≡ 5 (mod 6), u ≥ 53, Lemma 6.30 constructs the

ingredients for Lemma 6.49, some of which are confirmed to exist by Theorem 6.32. Hence,

we get the following theorem and its corollary. Note that when u ≤ 47, Lemma 6.30

provides a construction of these families of objects up to the existence of the auxiliary

objects (cf. Corollary 6.33).

Theorem 6.50. Let u0 ∈ {5, 11, 17, 23}.

1. Let g0 ∈ {2, 8, 14, 20}. If there exist an optimal, essential 4 − GDCD of type 8u0

and an optimal auxiliary 4 − GDCD of type g0u0, then there exists an optimal

4 − GDCD of type gu for all u ≥ 53, u ≡ u0 (mod 24), and g ≡ g0 (mod 24),

g ≥ 50.

2. Let g ∈ {26, 38, 44}. If there exists an optimal, auxiliary 4 − GDCD of type gu0,

then there exists an optimal 4−GDCD of type gu for all u ≡ u0 (mod 24), u ≥ 50.

Corollary 6.51. There exists an optimal 4−GDCD of type gu when one of the following

holds:

1. g ≡ 8 (mod 12) and u ≡ 5 (mod 24), except possibly when g = 44 or u = 29,

2. g ≡ 8 (mod 24) and u ≡ 11, 17 (mod 24), except possibly when u ∈ {35, 41},

3. g ≡ 2 (mod 24) and u ≡ 5, 11 (mod 24), except maybe when g = 26 or u ∈ {29, 35}.

Recall, Theorem 6.39 gives an optimal 4 − GDCD of type gu when g ≡ 5 (mod 6)

and u ≡ 2 (mod 6), except possibly when g = 11 or 17. When u ≥ 56 and g = 11 or 17,

we apply Theorem 6.49 using the ingredients given in Theorem 6.36 and the 4−GDCD

of type 526 constructed in Theorem 6.39.


Theorem 6.52.

1. Let g = 11. There exists an optimal 4−GDCD of type gu for all u ≡ 2 (mod 6),

except possibly when u ∈ {32, 38, 44, 50}.

2. Let g = 17. There exists an optimal 4 − GDCD of type gu for all u ≡ 8, 14, 20

(mod 24), except possibly when u ∈ {32, 38, 44}.

3. If there exists an optimal 4 − GDCD of type 1726, then there exists an optimal

4−GDCD of type 17u for all u ≡ 2 (mod 24), u ≥ 26, except possibly when u = 50.

Finally, we consider the case g, u ≡ 5 (mod 6). The following theorem follows from

Lemma 6.40 and Theorem 6.49 when u ≥ 53. If u ≤ 47, Corollary 6.41 gives a construction

of these families of GDCDs up to the existence of some ingredients.

Theorem 6.53. Let i ∈ {5, 11, 17, 23}.

1. Let g0 ∈ {5, 11, 17, 23} and u0 ∈ {5, 11, 17, 23}. If there exist an optimal essential

4 − GDCD of type 823 and an optimal, auxiliary 4 − GDCD of type g0u0, then

there exists an optimal 4−GDCD of type gu for all g ≡ g0 (mod 24), g ≥ 53, and

u ≡ u0 (mod 24), u ≥ 53.

2. Let g ∈ {29, 35, 41, 47} and u0 ∈ {5, 11, 17, 23}. If there exist an optimal essential

4−GDCD of type 8u and an optimal, auxiliary 4−GDCD of type gu0, then there

exists an optimal 4−GDCD of type gu for all u ≡ u0 (mod 24), u ≥ 53.

6.6 Another construction method when g ≡ 2 (mod 3)

and u ≡ 5 (mod 6)

In this section we consider another set of constructions of optimal 4−GDCDs of type

gu when g ≡ 2 (mod 3) and u ≡ 5 (mod 6). In this case, if there exists a 4 − GDCD


of type gu which meets the Schonheim lower bound, its excess graph is within a few

edges of a 1-regular graph. Corollaries 6.33 and 6.41 and Theorems 6.50 and 6.53 give

constructions of 4−GDCDs for g and u in these congruence classes. If we ignore small

values of g and u, and only consider g ≥ 50 and u ≥ 53, we require thirty two auxiliary

objects for our constructions; in particular we require optimal, auxiliary 4−GDCD of

type gu for any g ∈ {2, 5, 8, 11, 14, 17, 20, 23} and u ∈ {5, 11, 17, 23}. We have constructed

only six of these ingredients when (g, u) ∈ {(2, 5), (2, 11), (8, 5), (8, 11), (8, 17), (20, 5)}.

In this section, we give another construction for this family of 4 − GDCDs which

is similar to the constructions in Lemma 6.11 and Theorem 6.39. It depends on Lem-

mas 6.30 and 6.40, and it requires only auxiliary ingredients which have five groups.

However, it also requires essential ingredients whose excess graph is not regular. We start

by determining the lower bound on the size of a 4−GDCD of type g6(2g)1 when g ≡ 2

(mod 3), an essential ingredient in our new construction.

Lemma 6.54. Let g ≡ 2 (mod 3). A 4−GDCD of type g6(2g)1 has at least 4g2 + g(g+1)2

blocks. Moreover, if there exists a 4−GDCD of type g6(2g)1 having 4g2 + g(g+1)2

blocks,

it is optimal, and the vertices in its excess graph corresponding to the elements in the

groups of size g have degree one, and the vertices corresponding to the elements in the

group of size 2g have no incident edges.

Proof. Let V be the set of 8g elements. Partition V into six groups of size g and one

group X of size 2g. Any block containing an element x ∈ X has three elements from

V \X. Take all blocks containing an x ∈ X; there must be at least |V \X|3

= 6g3

= 2g of

them. Hence, there are at least (2g)2 = 4g2 blocks containing an element of X.

Let a ∈ V \X. For any x ∈ X, there exists a block containing both x and a. Hence,

a appears in at least 2g blocks containing an element of X. Therefore, in the subset of

blocks which contain an element of X, a appears in a block with at least 2g · 2 = 4g other

elements in V \X, and a has to appear in at least d5g−4g3e = dg

3e = g+1

3additional blocks,

since g ≡ 2 (mod 3). As this holds for any a ∈ X \V , there are at least 14(6g g+1

3) = g(g+1)

2


blocks not having an element in X.

If there exists a 4−GDCD of type g6(2g)1 having 4g2 + g(g+1)2

blocks, the structure

of its excess graph directly follows from the above computation.

When g ≡ 2 (mod 3), we say that there exists an optimal 4 − GDCD of type

g6(2g)1 if there exists a 4−GDCD of type g6(2g)1 which meets the lower bound given in

Lemma 6.54.

Lemma 6.55. Let g be a positive integer such that g ≡ 2 (mod 3). If there exist an

optimal, essential 4 − GDCD of type g6(2g)1 and an optimal, auxiliary 4 − GDCD of

type g5, then there exists an optimal 4−GDCD of type gu for all u ≡ 5 (mod 6), u ≥ 29.

Proof. Let u = 6l + 3 + 2, l ≥ 4. If g = 2, there exists a 4 − GDD of type 12l61 by

Theorem 2.10. Otherwise, we first construct a 4−GDD of type (6g)l(3g)1. Inflate each

point of a 4−GDD of type 6l31, which exists by Theorem 2.8, by g elements. For each

block of the 4−GDD of type 6l31, put a copy of a TD(4, g) onto corresponding elements.

Note, a TD(4, g) exists for all g 6∈ {2, 6} by Theorem 2.11.

Now, given a 4− GDD of type (6g)l(3g)1, add a set X of 2g infinite points. Put a

copy of an optimal 4−GDCD of type g6(2g)1 on the elements of each group of size 6g

and X such that the elements of the group of size 6g are partitioned into six subsets of

size g and X is kept as the group of size 2g. Finally, put a copy of an optimal 4−GDCD

of type g5 on the elements of the group of size 3g and X.

Together Lemmas 6.30 and 6.40 and Lemma 6.55 give the following theorem.

Theorem 6.56. If there exist an optimal, essential 4−GDCD of type g60(2g0)1 and an

optimal, auxiliary 4 − GDCD of type g50 for g0 ∈ {2, 5, 8, 11, 14, 17, 20, 23}, then there

exists an optimal 4−GDCD of type gu for all g = 32 or g ≡ g0 (mod 24), g ≥ 50, and

u ≡ 5 (mod 6), u ≥ 29.


Proof. Let u ≡ 5 (mod 6), u ≥ 29. By Lemma 6.55, there exist the optimal 4−GDCDs

of types 8u and gu0 . Then by Lemmas 6.30 and 6.40, there exists an optimal 4−GDCD

of type gu where g = 32 or g0 ≡ g (mod 24), g ≥ 50.

The cases omitted by Theorem 6.56 are:

• u ∈ {11, 17, 23} and g = 32 or g ≥ 50, considered by Lemmas 6.30 and 6.40, and

• g ∈ {26, 29, 35, 38, 41, 44, 47} and u ≥ 53, regarded by Theorems 6.50 and 6.53.

If we ignore these infinite families of 4−GDCD with small values of g or u, Theo-

rem 6.56 gives an optimal solution for this congruence class which requires only sixteen

ingredients which is a half of the number of ingredients required by Theorems 6.50 and 6.53.

However, for an application of Theorem 6.56, a half of the ingredients are essential and

their excess graph is not regular. We have been able to construct only two of them for

g = 2 (Lemma 6.9) and g = 5 (Lemma 6.38). On the other hand, we only miss one

essential ingredient in the application of Theorems 6.50 and 6.53, and it is a 4−GDCD

of type 823. Therefore, Theorems 6.50 and 6.53 imply a stronger result.

6.7 Summary of results

In Sections 6.4 and 6.5, we give constructions of a 4 − GDCD of type gu using two

different approaches: in Section 6.4, for a fixed value of u, we consider congruence classes

of g (mod 12); in Section 6.5, for a fixed value of g, we study constructions for different

congruence classes of u (mod 12). Together they reduce the problem of the existence of

an optimal 4−GDCD of type gu to finding finitely many ingredient 4−GDCDs whose

size meets the Schonheim lower bound. Since we have found an optimal solution for

almost all essential ingredients, the size of the resulting families of 4−GDCDs may differ

from the lower bound by at most a constant number of blocks.

Moreover, we have constructed many optimal families of 4−GDCDs. We summarize

our results in the following theorems, first of which considers the 4 − GDCDs with a


regular excess graph, and the second regards the optimal 4−GDCDs whose excess graph

is not regular.


⌈g(u−1)

3

⌉⌉when u ≥ 4 and one of the following holds:

1. g ≡ 0 (mod 6) except when (g, u) = (6, 4),

2. g ≡ 3 (mod 6) and u ≡ 0, 1 (mod 4),

3. g ≡ 2, 4 (mod 6) and u ≡ 7, 10 (mod 12),

4. g ≡ 1, 2, 4, 5 (mod 6) and u ≡ 1, 4 (mod 12), except when (g, u) = (2, 4),

5. g ≡ 1, 4 (mod 6) and u ≡ 0, 6, 8, 11 (mod 12),

6. g ≡ 4 (mod 12) and u ≡ 3 (mod 6), except possibly when u ∈ {21, 27},

7. g ≡ 4 (mod 6) and u ≡ 2, 5 (mod 12),

8. g ≡ 2 (mod 6) and u ≡ 0, 2, 3 (mod 6),

9. g ≡ 8 (mod 12) and u ≡ 5 (mod 24), except possibly when g = 44 or u = 29,

10. g ≡ 8 (mod 24) and u ≡ 11, 17 (mod 24), except possibly when u ∈ {35, 41},

11. g ≡ 5 (mod 6), and u ≡ 2 (mod 6), except possibly when g ∈ {11, 17} and u ∈

{32, 38, 44, 50}, or g = 17 and u ≡ 2 (mod 24), u ≥ 26,

12. g ≡ 5 (mod 6) and u ≡ 0, 3 (mod 12), except possibly when g = 17, or u ∈

{27, 36, 39, 51}, or (g, u) = (11, 24).


⌈g(u−1)

3


1. g ≡ 1 (mod 12) and u ≡ 2, 3, 5, 9 (mod 12), except when (g, u) = (1, 9) and possibly

when g = 13, or u = 9 and g ≥ 13,


2. g ≡ 2 (mod 24) and u ≡ 5, 11 (mod 24), except possibly when g = 26 or u ∈

{29, 35},

3. g = 2 and u ≡ 17, 23 (mod 24), except possibly when u ∈ {17, 23},

4. g ≡ 3 (mod 6) and u ≡ 2, 3 (mod 4), except when (g, u) = (3, 6) and possibly when

(g, u) ∈ {(15, 14), (21, 14), (15, 18), (21, 18)}, or u = 6 and g ≥ 9,

5. g ≡ 1, 5 (mod 6) and u ≡ 7, 10 (mod 4), except when g = 1 and u ∈ {7, 10, 19},

and possibly when g ∈ {5, 7}.

We can also represent the results of Theorems 6.57 and 6.58 with respect to congruence

classes u (mod 12).


⌈g(u−1)

3


1. u ≡ 0 (mod 12) except possibly when g = 17, or when u = 36 and g ≡ 5 (mod 6),

or when (g, u) = (11, 24);

2. u ≡ 1, 4 (mod 12), except when (g, u) ∈ {(2, 4), (6, 4)};

3. u ≡ 2 (mod 12), except possibly when g = 13, or g ≡ 7 (mod 12), or g = 17 and

u ≡ 2 (mod 24), or (g, u) ∈ {(15, 14), (21, 14), (11, 38), (17, 38)};

4. u ≡ 3 (mod 12), except possibly when g ∈ {13, 17}, or g ≡ 7, 10 (mod 12), or

u ∈ {27, 39, 51} and g ≡ 5 (mod 6), or u = 27 and g ≡ 4 (mod 12);

5. u ≡ 5 (mod 12), except possibly when g = {13, 26, 44}, or u ∈ {29, 41} and g ≡ 2, 8

(mod 24), or g ≡ 5 (mod 6), or g ≡ 14 (mod 24), or u ≡ 17 (mod 24) and g ≡ 20

(mod 24), or u ≡ 17 (mod 24) and g ≡ 2 (mod 24);

6. u ≡ 6 (mod 12), except when (g, u) = (3, 6), and possibly when u = 6 and g ≡ 3

(mod 6), g ≥ 9, or g ≡ 5 (mod 6), or (g, u) ∈ {(15, 18), (21, 18)};

7. u ≡ 7 (mod 12), except when (g, u) ∈ {(1, 7), (1, 19)}, and possibly when g ∈ {5, 7};


8. u ≡ 8 (mod 12), except possibly when (g, u) ∈ {(11, 32), (11, 44), (17, 32), (17, 44)};

9. u ≡ 9 (mod 12), except when (g, u) = (1, 9), and possibly when g = 13, or g ≡

5, 7, 10, 11 (mod 12), or u = 9 and g ≡ 1 (mod 12), g ≥ 13, or u = 21 and g ≡ 4

(mod 12);

10. u ≡ 10 (mod 12), except when (g, u) = (1, 10), and possibly when g ∈ {5, 7};

11. u ≡ 11 (mod 12), except possibly when g = 26, or u = 35 and g ≡ 2, 8 (mod 24),

or g ≡ 5 (mod 6), or u ≡ 11 (mod 24) and g ≡ 14, 20 (mod 24), or u ≡ 23

(mod 24) and g ≡ 2 (mod 6), g ≥ 8, or (g, u) = (2, 23).

In addition, we constructed two families of close to optimal 4−GDCDs which illustrate

the concept of auxiliary ingredients in the constructions. The following theorem is a

consequence of the fact that the optimal sizes of a (9, 4, 1)-covering design and a 4−GDCD

of type 36 do not meet the Schonheim lower bound.

Theorem 6.60.

1. When g ≡ 3 (mod 6), there exists a close to optimal 4 − GDCD of type g6 with⌈3g2

⌈5g3

⌉⌉+ 2 blocks, which is optimal when g = 3.

2. When g ≡ 1 (mod 12), g 6= 13, there exists a close to optimal 4−GDCD of type

g9 having⌈9g4

⌈8g3

⌉⌉+ 1 blocks, which is optimal when g = 1.

The constructions in Sections 6.4 and 6.5 have the potential to completely solve all

other cases provided that an optimal solution for the auxiliary ingredients meets the

Schonheim lower bound. However, these ingredient 4 − GDCDs have a large number

of blocks and cannot be obtained using our computer search. For a summarized list of

the constructions, see Section 6.8. Appendix A contains lists of ingredients which are

necessary for the constructions but their optimal size is undetermined. It also contains

the list of individual cases of 4−GDCD which are exceptions for all of our constructions.


All of these ingredients other than a 4−GDCD of type 1712 and a 4−GDCD of type

823 are auxiliary. Also, we need an essential 4−GDCD of type 8u when u ∈ {29, 35, 41, 47},

but these essential ingredients are only necessary for constructions of 4−GDCD with

these individual values of u. Therefore, we get the statement of our main Theorem 6.2

for which the exceptions are given by the unknown optimal size for these six essential

ingredients. For all other cases, we have constructed essential ingredients, hence their size

only depends on the size of the auxiliary ingredients. The constant δ in Theorem 6.2 is

equal to the maximum number of blocks for which an auxiliary ingredient exceeds the

Schonheim lower bound. The maximum exists since there are finitely many auxiliary

ingredients necessary for our constructions in Sections 6.4 and 6.5. At the beginning of

this chapter, we conjectured that all missing ingredients meet the Schonhim lower bound

(cf. Conjecture 6.3).

6.8 Summary of constructions

Table 6.2 summarizes the constructions presented in this chapter. It also contains a list

of the ingredient objects for which optimal solutions are still undetermined. The rows

and columns in the table represent congruence classes: rows are classes u (mod 12), and

columns are classes g (mod 6). When g ≡ 0 (mod 6), a 4−GDCD of type gu is a group

divisible design for any u ≥ 4, hence the column zero is omitted.

For example, when u ≡ 5 (mod 12) and g ≡ 5 (mod 6), the first line in this cell

means that for a fixed u, if there exist an optimal, essential 4−GDCD of type 8u and the

optimal, auxiliary 4−GDCDs of types 5u, 11u, 17u, and 23u, then there exists an optimal

4−GDCD of type gu for all g ≥ 53 (Lemma 6.40). The second line says that for a fixed g,

if there exist an essential 4−GDCD of type g8 and the auxiliary 4−GDCDs of types g5

and g17, then there exists an optimal 4−GDCD of type gu for all u ≥ 53 (Lemma 6.49).

The last two lines indicate that in order to apply these two constructions, we still need to


find an optimal 4−GDCD of type gu for all g ∈ {5, 11, 17, 23} and u ∈ {5, 17}.


Tab

le6.

2:Sum

mar

yof

const

ruct

ions.

Leg

end:

M=

ingr

edie

nts

requir

edfo

ra

const

ruct

ion,

but

not

know

nto

mee

tth

eSch

onhei

mlo

wer

bou

nd,

E=

exce

pti

onin

aco

nst

ruct

ion,

*=

esse

nti

alob

ject

,u

(mod

12),

andg

(mod

6).

u�g

12

34

5

0g=

1∗⇒g≥

7g=

2∗⇒g≥

8g=

1∗⇒g≥

4g=

2∗,5⇒g≥

23

u=

12∗⇒u≥

48

M:g=

17,u

=12

1 2

g=

1,7⇒g≥

25

g=

2∗⇒g≥

8

g=

3⇒g≥

9g=

4,1

0⇒g=

16,g≥

28

g=

2∗,5⇒g≥

23

u=

5⇒u≥

17

E:g=

15,2

1,u

=14

u=

5⇒u≥

17

56101⇒g=

5,u≥

26

M:g=

7,1

3,1

9,u

=5

u=

8∗,1

4,2

6⇒u≥

62

M:(g,u

)=

(17,2

6)

3g=

1,7⇒g≥

25

g=

2∗⇒g≥

8g=

3⇒g≥

9g=

4,1

0⇒g=

16,g≥

28

g=

5⇒g≥

23

u=

6∗,9⇒u≥

39

u=

6∗,9⇒u≥

39

u=

12∗,1

5⇒u≥

63

M:g=

7,u

=9

M:g=

10,u

=9

M:g=

17,u

=12

4 5

g=

1,7⇒g≥

25

g=

2,8∗,1

4,2

0⇒g=

32,g≥

50

g=

4,1

0⇒g=

16,g≥

28

g=

8∗,5,1

1,1

7,2

3⇒g≥

53

u=

5⇒u≥

17

u=

8∗,5,1

7⇒u≥

53

u=

5⇒u≥

17

u=

8∗,5,1

7⇒u≥

53

M:g=

7,1

3,1

9,u

=5

M:g=

14,2

0,2

6,3

8,4

4M:g=

5,1

1,1

7,2

3u=

5,1

7,(g,u

)=

(2,1

7)

u=

5,1

7

6g=

1∗⇒g≥

7g=

2∗⇒g≥

8

g=

3⇒g≥

9

g=

1∗⇒g≥

4

g=

2∗,5⇒g≥

23

E:g=

15,2

1,u

=18

u=

12∗,6⇒u≥

54

M:g=

3,u

=6

M:(g,u

)=

(17,1

2),(5,6

)M:(g,u

)=

(11,6

),(17,6

)

7g=

3⇒g≥

13

g=

3⇒g≥

9g=

3⇒g≥

11

u=

7⇒u≥

31

u=

7⇒u≥

31

u=

7⇒u≥

31

M:g=

7,u

=7

M:g=

5,u

=7

8g=

1∗⇒g≥

7g=

2∗⇒g≥

8g=

1∗⇒g≥

4g=

2∗,5⇒g≥

23

56101⇒g=

5,u≥

26

u=

8∗,2

0⇒u≥

56

9

g=

1,7⇒g≥

25

g=

2∗⇒g≥

8

g=

4,1

0⇒g=

16,g≥

28

g=

5⇒g≥

23

u=

6∗,9⇒u≥

33

u=

6∗,9⇒u≥

33

u=

12∗,9⇒u≥

57

M:g=

7,u

=9

M:g=

10,u

=9

M:(g,u

)=

(17,1

2),(5,9

)M:(g,u

)=

(11,9

),(17,9

)

10

g=

3⇒g≥

13

g=

3⇒g≥

9g=

3⇒g≥

11

u=

7⇒u≥

22

u=

7⇒u≥

31

u=

7⇒u≥

22

M:g=

7,u

=7

M:g=

5,u

=7

11

g=

1∗⇒g≥

7

g=

2,8∗,1

4,2

0⇒g=

32,g≥

50

g=

3⇒g≥

9

g=

1∗⇒g≥

4

g=

8∗,5,1

1,1

7,2

3⇒g≥

53

u=

8∗,1

1,2

3⇒u≥

59

u=

8∗,1

1,2

3⇒u≥

59

M:g=

2,1

4,2

0,2

6,3

8,4

4M:g=

5,1

1,1

7,2

3u=

11,2

3,(g,u

)=

(8,2

3)

u=

11,2

3

Chapter 7

Packing arrays with row limit with

constant block size

Packing arrays with row limit, PARLs, are another generalization ofGDDs and orthogonal

arrays. The definition of a PARL is almost the same as the definition of a CARL with

the difference that each t-tuple is covered at most λ times, where λ is the index (see

Definition 3.5). Moreover same as a CARL, a PARL has three representations: as

an array, as a triple of sets, and as a graph packing problem, which we introduced in

Chapter 3.

Some of the results on CARLs in Chapters 4 and 6 extend nicely to PARLs. In

particular, analogous to the Schonheim lower bound, we derive an upper bound for

PARLs, which we call the Schonheim upper bound (cf. Theorem 7.1). When the row

limit is a constant, we show that the optimal size of PARLs is asymptotically equal to

the Schonheim upper bound as a function of k, the number of columns (see Theorem 7.2).

Moreover, the proof of this theorem suggests that removing a set of rows of an optimal

CARL which contribute to the edges of the excess graph gives a close to optimal PARL.

We apply this idea to construct optimal PARLs with row limit three and strength two

from optimal CARLs. More precisely, in Section 7.2 we construct optimal 3−GDCDs

120

Chapter 7. Packing arrays with row limit with constant block size 121

which transform into optimal 3 − GDPDs. However, before we start this study, we

derive a generalization of the Johnson upper bound for packings, which strengthens the

Schonheim upper bound for two families of 3−GDPDs. We also give an analogue of the

Johnson bound for CARLs. We finalize this chapter by showing that optimal 3−GDPDs

meet the Schonheim and Johnson upper bounds, with only eight possible exceptions (see

Theorem 3.1).

7.1 Upper Bounds

We start by deriving an upper bound for PARLs which is analogous to the Schonheim

lower bound for CARLs. Moreover, we show that it is asymptotically equal to the optimal

size of a PARL with a constant row limit. Next we generalize the Johnson bound for

packings in the context of PARLs in order to strengthen the Schonheim bound for two

classes of PARL with row limit three, which we construct in Section 7.2.

7.1.1 Schonheim upper bound

Analogously to the Schonheim lower bound for CARLs, we can derive an upper bound

on the size of a PARL. The idea is to take a subarray with a fixed alphabet element in

one column. This subarray is an embedded PARL with smaller strength. Repeating the

process recursively we get a subarray which is a PARL with strength t = 1 for which one

can easily construct. The only difference is that we get an upper bound, and the ceiling

functions are replaced by floor functions.

Theorem 7.1.

PARLNλ(t, k, v : w) ≤ UBλ(t, k, v : w)


where

UBλ(t, k, v : w) =

⌊vk

w

⌊v(k − 1)

w − 1

⌊· · ·⌊v(k − t+ 2)

w − t+ 2

⌊λv(k − t+ 1)

w − t+ 1

⌋⌋· · ·⌋⌋⌋

.

Proof. Let A = (aij) be a PARLλ(N ; t, k, v : w). Let the alphabet set be V , |V | = v.

We count the number of non-empty cells in A in two ways. First, by rows, there are

Nw non-empty cells in A. Alternatively, if we denote by r(xc) the number of times an

element x ∈ V appears a in column c ∈ [1, k], then counting by columns, the number of

non-empty cells is

Nw =k∑c=1

∑x∈V

r(xc) ≤ kvr(xc00 ),

where x0 the element of the alphabet in column c0 which occurs the most in A, that is,

r(xc00 ) = max{r(xc) : c ∈ [1, k], x ∈ V }. On the other hand, the subarray of A which we

get by taking the rows of A which contain x0 in the column c0 and then deleting the column

c0 is a PARLλ(t− 1, k − 1, v : w − 1), and so PARLNλ(t− 1, k − 1, v : w − 1) ≥ r(xc00 ).

Hence,

(7.1) PARLNλ(t, k, v : w) ≤ N ≤⌊kv

wPARLNλ(t− 1, k − 1, v : w − 1)

⌋.

We can iterate the inequality (7.1) till t = 1. It is easy to see that by cycling through

columns and alphabet signs, we get a PARLλ(N′; 1, k − t + 1, v : w − t + 1) of size

N ′ =⌊λv(k−t+1)w−t+1

⌋. Therefore, we get

PARLNλ(t, k, v : w) ≤⌊vk

w

⌊v(k − 1)

w − 1

⌊· · ·⌊v(k − t+ 2)

w − t+ 2

⌊λv(k − t+ 1)

w − t+ 1

⌋⌋· · ·⌋⌋⌋

.


7.1.2 Asymptotic size of PARLs with constant row limit

Theorem 4.7 gives the asymptotic size of the family of CARLs for which the row limit

is a constant. If we delete the rows of such a CARL which contribute to the excess, we

get a PARL of asymptotically equal size. Rodl [46] made an analogous observation for

coverings and packings.

Theorem 7.2. Let t, k, v and w be positive integers such that t ≤ w ≤ k. Then,

PARLN(t, k, v : w) =

(kt

)(wt

)vt(1 + o(1)) as k →∞.

Proof. By Theorem 4.7, for any ε > 0, there exists k0 such that for all positive integers

k ≥ k0, there exists a CARL(N ; t, k, v : w), A, such that N ≤ (kt)

(wt)vt(

1 + ε(wt)−1

), since

w and t are constants.

Denote by rA(τ) be the number of rows of A which cover the t-tuple τ ∈ T . For

any τ ∈ T , rA(τ) ≥ 1. Let E be the minimum number of rows of A which we need to

remove such that in the truncated array, A′, rA′(τ) ≤ 1 for all τ ∈ T , that is, A′ is a

PARL. Then, E is smaller than or equal to the number of edges in the excess graph

of A, i.e. E ≤ N(wt

)−(kt

)vt. Indeed, since each row covers

(wt

)t-tuples, there are N

(wt

)t-tuples covered in A in total, and all over

(kt

)vt correspond to an edge in the excess


graph. Therefore,

PARLN(t, k, v : w) ≥ N − E ≥ N

(1−

(w

t

))+

(k

t

)vt

=

(k

t

)vt −

((w

t

)− 1

)N

≥(k

t

)vt −

((w

t

)− 1

) (kt

)(wt

)vt(1 +ε(

wt

)− 1

)

=

(k

t

)vt −

(k

t

)vt

(1 +

ε(wt

)− 1

)+

(kt

)(wt

)vt(1 +ε(

wt

)− 1

)

=

(kt

)(wt

)vt(− ε(wt

)(wt

)− 1

+ 1 +ε(

wt

)− 1

)

=

(kt

)(wt

)vt(1− ε).On the other hand, the Schonheim upper bound in Theorem 7.1 satisfies the following

inequality:

UB(t, k, v : w) ≤(kt

)(wt

)vt.Hence, the lower and the upper bounds have the same size asymptotically.

The proof of Theorem 2.26, on which Theorem 4.7 is based, is constructive. This

means that with high probability, one can construct a CARL with which we start the

proof of Theorem 7.1. Then deleting the rows which contribute the excess, we construct

a PARL whose size is asymptotically optimal.

7.1.3 The Johnson bounds for PARLs and CARLs

The Schonheim upper bound in Theorem 7.1 is obtained by counting and the fact that

a subarray of a PARL obtained by fixing a value of the alphabet in a single column is

again a PARL. We can further refine this computation by taking a subarray of a PARL

with a fixed m-tuple of alphabet elements across m columns instead of a single element.

For packings, PARLs with v = 1, such bound is called the first Johnson bound [29].


Theorem 7.3. If vk PARLNλ(t− 1, k− 1, v : w− 1) 6≡ 0 (mod w) and for some m such

that 2 ≤ m ≤ t,

PARLNλ(t− 1, k − 1, v : w − 1) =

(k−1m−1

)(w−1m−1

)vm−1PARLNλ(t−m, k −m, v : w −m),

then

PARLNλ(t, k, v : w) ≤⌊vkPARLNλ(t− 1, k − 1, v : w − 1)−m

w

⌋.

Proof. Given positive integers t and k, and a v-set V , let anm-tuple µ = {xc11 , xc22 , . . . , xcmm }

be an assignment of the alphabet symbol xi ∈ V to the column ci for i ∈ [1,m]. Let Tm

be the set of all m-tuples which may be covered in a PARL. Then |Tm| =(km

)vm. Let A

be a optimal PARL(N ; t, k, v : w), and denote by r(µ) the number of rows of A which

cover an m-tuple µ.

Given an m-tuple µ, the subarray of A obtained by taking only the rows of A which

cover µ, and then deleting the m columns which contain µ, is a PARLλ(t − m, k −

m, v : w −m), and hence PARLNλ(t−m, k −m, v : w −m) ≥ r(µ).

Let W = {(ρ, µ) : ρ is a row of A which covers an m-tuple µ ∈ Tm}. We compute

the order of W .

If for every µ ∈ Tm, r(µ) = PARLNλ(t−m, k −m, v : w −m), then

|W | = N

(w

m

)=∑µ∈Tm

r(µ) =

(k

m

)vmPARLNλ(t−m, k −m, v : w −m).

Therefore,

PARLNλ(t, k, v : w) = N =

(km

)(wm

)vmPARLNλ(t−m, k −m, v : w −m),


that is

wPARLNλ(t, k, v : w) = kv

(k−1m−1

)(w−1m−1

)vm−1PARLNλ(t−m, k −m, v : w −m) =

= kvPARLNλ(t− 1, k − 1, v : w − 1),

which contradicts the congruency condition of the theorem.

Therefore, there exists a µ0 ∈ Tm, such that r(µ0) < PARLNλ(t−m, k−m, v : w−m).

For every xcii ∈ µ0, i ∈ [1,m], define

W (xcii ) = {(ρ, µ) : µ ∈ Tm such that xcii ∈ µ and ρ is a row of A which covers µ}.

Then,

|W (xcii )| = r(xcii )

(w − 1

m− 1

)=∑µ∈Tmxcii ∈µ

r(µ) <

(k − 1

m− 1

)vm−1PARLNλ(t−m, k−m, v : w−m).

Therefore,

r(xcii ) <

(k−1m−1

)(w−1m−1

)vm−1PARLNλ(t−m, k −m, v : w −m)

= PARLNλ(t− 1, k − 1, v : w − 1),

that is

r(xcii ) ≤ PARLNλ(t− 1, k − 1, v : w − 1)− 1.

Finally, we redo the counting in the proof of Theorem 7.1. The number of non-empty

cells in A equals:

Nw =∑x∈Vc∈[1,k]

r(xc) ≤ vkPARLNλ(t− 1, k − 1, v : w − 1)−m,


since for any x ∈ V and c ∈ [1, k], r(xc) ≤ PARLNλ(t−1, k−1, v : w−1) and m elements

belong to µ0. Hence,

PARLNλ(t, k, v : w) = N ≤⌊vkPARLNλ(t− 1, k − 1, v : w − 1)−m

w

⌋

In particular, for t = 2, we are interested when the Johnson bound of Theorem 7.3 is

an improvement to the Schonheim upper bound of Theorem 7.1.

Corollary 7.4. If λ, v, w and k are given positive integers such that 2 ≤ w ≤ k, and

λv(k − 1) ≡ 0 (mod w − 1) and λv2k(k − 1) ≡ −1 (mod w), then

PARLNλ(2, k, v : w) ≤⌊λv2k(k − 1)

w(w − 1)

⌋− 1.

Proof. By Theorem 7.1, PARLNλ(2, k, v : w) ≤⌊vkw

⌊λv(k−1)w−1

⌋⌋. On the other hand,

Theorem 7.3 implies that when m = 2, if⌊λv(k−1)w−1

⌋= λv(k−1)

w−1 , and vk λv(k−1)w−1 6≡ 0 (mod w),

then PARLλ(2, k, v : w) ≤⌊vkw

⌊λv(k−1)w−1

⌋− 2

w

⌋. Hence, when λv(k− 1) ≡ 0 (mod w− 1),

⌊λv2k(k − 1)

w(w − 1)− 2

w

⌋=

⌊λv2k(k − 1)

w(w − 1)

⌋− 1,

if and only if vk λv(k−1)w−1 ≡ 0, 1 (mod w). Since a condition of the Johnson bound is that

vk λv(k−1)w−1 6≡ 0 (mod w), we must have that vk λv(k−1)

w−1 ≡ 1 (mod w), that is λv2k(k−1) ≡

−1 (mod w).

Johnson lower bound for CARLs

Analogously, we can derive the Johnson lower bound and its corollary for CARLs.

The steps in the computations are the same, with the difference that a subarray of a

CARL(t, k, v : w) obtained by taking the subset of rows with a fixed m-tuple in a set


of m columns and then deleting these m columns, has size greater than or equal to

CARLN(t−m, k −m, v : w −m).

Theorem 7.5. If vk CARLNλ(t− 1, k− 1, v : w− 1) 6≡ 0 (mod w) and for some m such

that 2 ≤ m ≤ t,

CARLNλ(t− 1, k − 1, v : w − 1) =

(k−1m−1

)(w−1m−1

)vm−1CARLNλ(t−m, k −m, v : w −m),

then

CARLNλ(t, k, v : w) ≥⌈vkCARLNλ(t− 1, k − 1, v : w − 1) +m

w

⌉.

Corollary 7.6. If λ, v, w and k are given positive integers such that 2 ≤ w ≤ k, and

λv(k − 1) ≡ 0 (mod w − 1) and λv2k(k − 1) ≡ 1 (mod w), then

CARLNλ(2, k, v : w) ≥⌊λv2k(k − 1)

w(w − 1)

⌋+ 1.

The conditions of Corollary 7.6 are not satisfied when t = 2, λ = 1 and w = 4, hence

it does not affect the results in Chapter 6. It also does not change the asymptotic results

discussed in Section 4.2.3.

7.2 Construction of optimal 3−GDPDs from optimal

3−GDCDs

The proof of Theorem 7.2 suggests that, when the row limit is a constant, we can construct

an optimal PARL from an optimal CARL by removing a set of rows which contribute

to the edges of the excess graph of the CARL. The relation between optimal covering

designs and optimal packing designs was briefly explored before in [37]. Removing rows

which contribute excess is the first step in our construction of optimal PARLs from

CARLs with t = 2 and w = 3, which is our objective in this section. Given that the


strength is t = 2 and the row limit is a constant, we are going to construct optimal group

divisible packing designs with block size 3 from optimal group divisible covering designs

with the same block size. By removing the blocks from an optimal 3 − GDCD which

introduce edges to the excess graph, we obtain a 3−GDPD which may have a big leave

since we removed too many pairs which were covered only once. To add new blocks to the

packing, the obtained intermediate leave graph must contain sufficient number of edge

independent triangles, K3’s, which correspond the new blocks to be added to the packing.

From now on, we use the notation for GDPDs. Recall, the maximal (optimal) size of

a k−GDPD of type gu is denoted by D(k, gu) (cf. Definition 3.8.) Theorems 7.1 and 7.3

directly imply the following corollary.

Corollary 7.7. Let g and u ≥ 3 be positive integers. Then, D(3, gu) ≤ U(3, gu), where

U(3, gu) =

⌊gu

3

⌊g(u− 1)

2

⌋⌋− δ,

and

δ =

1, u ≡ 2 (mod 6) and g ≡ 2, 4 (mod 6),

1, u ≡ 5 (mod 6) and g 6≡ 0 (mod 3),

0, otherwise.

For positive integers g and u ≥ 3, the cover number for a 3 − GDCD of type gu is

known: C(3, gu) =⌈gu3

⌈g(u−2)

2

⌉⌉(cf. Theorem 3.1). As in Section 6.2, we can compute

the degree sequence and the number of edges in the leave graph of a 3−GDPD of type

gu having U(3, gu) blocks. Table 7.1 gives the number of edges in the leave graph of such

a GDPD. Table 7.2 does the same for the optimal 3−GDCDs.

Our goal is to construct a 3−GDPD of type gu which meets the upper bound U(3, gu)

for any positive integers g and u ≥ 3. Moreover, we want to do this by transforming an

optimal 3 − GDCD into the required 3 − GDPD. Denote by E(k, gu) the number of

edges in the excess graph of an optimal k −GDCD of type gu. The value of E(3, gu) is


u\g 0 1 2 3 4 5

0 0 gu2† 0 gu

2† 0 gu

2†

1 0 0 0 0 0 0

2 0 gu2† 4 gu

2† 4 gu

2+ 2

3 0 0 0 0 0 0

4 0 gu2

+ 1 0 gu2† 0 gu

2+ 2

5 0 4 4 0 4 4

Table 7.1: The number of edges in the leave graph of a 3−GDPD of type gu with U(3, gu)blocks, if it exists. A (†) denotes a 1-regular leave graph.

u\g 0 1 2 3 4 5

0 0 gu2† 0 gu

2† 0 gu

2†

1 0 0 0 0 0 0

2 0 gu2

+ 1 2 gu2† 2 gu

2†

3 0 0 0 0 0 0

4 0 gu2

+ 1 0 gu2† 0 gu

2+ 2

5 0 2 2 0 2 2

Table 7.2: The number of edges in the excess graph of an optimal 3−GDCD of type gu.A (†) denotes a 1-regular excess graph.

given in Table 7.2. Then, the transformation process is as follows: (1) remove E(3, gu)

blocks from an optimal GDCD to get a GDPD, (2) if the obtained GDPD does not

meet the upper bound U(3, gu), add blocks to get an optimal GDPD. We formalize this

process in Definition 7.1.

Definition 7.1. Let g, k, u be positive integers such that u ≥ k. An optimal k−GDCD

of type gu, (V,G,Bc), transforms into an optimal k −GDPD of type gu, (V,G,Bp), if (1)

there exists a subset B1 ⊂ Bc such that |B1| = E(k, gu) and (2) there exists a collection

of k-subsets of V , B2, such that Bp = (Bc \ B1) ∪ B2.

We say that a optimal GDCD is transferable if it transforms into an optimal GDPD.

We present constructions of the optimal 3 − GDCDs similar to the constructions

of the 4 − GDCDs in Chapter 6, such that the ingredients in the constructions can

be transformed from a covering to a packing. We refer our reader to the summary of

the constructions in Subsection 7.2.8 which may help the reader to more easily follow


the constructions. Since we want to construct a family of optimal 3−GDPDs for any

value of g and u, we restrict our attention to the two step process of constructing the

transformable GDCD as was done in Chapter 6: for a fixed number of groups u, we

construct a family of 3 − GDCDs of type gu for some congruence class of g (mod 6)

by filling holes of an appropriate IGDD or a DGDD; then for a fixed group size g, we

construct the family of 3−GDCDs of type gu for some congruence class of u (mod 6)

using Wilson’s construction. Every application of Wilson’s Construction 6.4 requires an

essential 3−GDCD of type g3, which is in fact a GDD since its excess graph is empty,

and it exists for any positive integer g by Theorem 2.3. We do not emphasize this fact

later in the applications of Wilson’s construction.

7.2.1 Examples of optimal 3−GDCDs which transform into op-

timal 3−GDPDs

We start with examples of optimal 3 − GDCDs of type gu for some fixed values of g

and u which transform into optimal 3 − GDPDs. Later, we use these 3 − GDCDs as

ingredients in the constructions in Subsections 7.2.3-7.2.5.

Lemma 7.8. Let (g, u) ∈ {(1, 4), (1, 6), (1, 8), (3, 4)}. There exists an optimal 3−GDCD

of type gu, which transforms into an optimal 3−GDCD of type gu with U(3, gu) blocks.

Proof. Let Zgu be the set of elements, and let group Gi = {x ∈ Zgu : x ≡ i (mod u)},

i ∈ Zu. Following Definition 7.1, for each pair (g, u), we give the collections of blocks Bc,

B1 and B2 of an optimal GDCD which transforms into an optimal GDPD.

• (g, u) = (1, 4)

Blocks of a GDCD, Bc = {{0, 1, 2}, {0, 2, 3}, {0, 1, 3}}

Blocks to be deleted, B1 = {{0, 2, 3}, {0, 1, 3}}

There are no blocks to be added, B2 = {∅}.

• (g, u) = (1, 6)


Blocks of a GDCD, Bc:

{0, 1, 2}

{1, 2, 5}

{0, 3, 5}

{0, 4, 5}

{1, 3, 4}

{2, 3, 4}

Blocks to be deleted, B1 = {{0, 1, 2}, {0, 4, 5}, {2, 3, 4}}

Block to be added, B2 = {{1, 3, 5}}.

• (g, u) = (1, 8)


{2, 5, 6}

{3, 4, 5}

{2, 3, 7}

{0, 4, 7}

{1, 4, 6}

{3, 6, 8}

{1, 5, 7}

{1, 3, 2}

{0, 2, 4}

{0, 1, 5}

{5, 6, 7}

Blocks to be deleted, B1:

{1, 3, 2}

{0, 2, 4}

{0, 1, 5} {5, 6, 7}

Block to be added, B2 = {{0, 1, 2}}.

• (g, u) = (3, 4):



{3, 5, 0}

{2, 3, 8}

{7, 5, 2}

{9, 0, 10}

{5, 10, 4}

{9, 4, 7}

{7, 10, 1}

{6, 4, 3}

{11, 9, 4}

{6, 8, 1}

{11, 10, 8}

{1, 4, 2}

{8, 6, 9}

{11, 2, 0}

{2, 9, 3}

{1, 10, 3}

{7, 5, 8}

{7, 6, 0}

{6, 5, 11}

{1, 0, 11}

Blocks to be deleted, B1:

{7, 10, 1}

{11, 9, 4}

{6, 8, 1}

{11, 2, 0}

{2, 9, 3}

{7, 5, 8}

Blocks to be added, B2 = {{1, 7, 8}, {2, 9, 11}}

Lemma 7.9. There exists an optimal 3−GDCD of type 112, which transforms into an

optimal 3−GDCD of type 112.

Proof. Let Z12 be the set of elements, and let group Gi = {i} for all i ∈ Z12. We give

three collections of blocks: starters blocks of a GDCD, starter blocks to be deleted, and

blocks to be added but not to be developed. To obtain blocks of a 3−GDCD of type 112,

develop the starter blocks +4 (mod 12). To transform the 3−GDCD to a 3−GDPD of

type 112, remove the starter rows to be deleted and all their developments +4 (mod 12)

and adjoin the blocks to be added. The blocks to be added should not be developed.


Starter blocks of a GDCD:

{1, 11, 8}

{2, 8, 7}

{3, 1, 7}

{2, 9, 0}

{2, 5, 4}

{2, 1, 5}

{2, 10, 11}

{0, 11, 4}

Starter blocks to be deleted:{{2, 1, 5}, {0, 11, 4}}

Blocks to be added:{{0, 4, 8}, {1, 5, 9}}

Lemma 7.10. Let (g, u) ∈ {(3, 8), (3, 10)}. There exists an optimal 3−GDCD of type

gu, which transforms into an optimal 3−GDCD of type gu.

Proof. Let Zgu be the set of elements, and the groups Gi = {x ∈ Zgu : x ≡ i (mod u)} for

all i ∈ Zu. For each pair (g, u), we give a positive integer d, and three collections of blocks:

starters blocks of a GDCD, starter blocks to be deleted, and blocks to be added. To

obtain blocks of a 3−GDCD of type gu, develop the starter blocks +d (mod gu). Since

the edges in the excess graph come from the development of the short cycle, to transform

the 3−GDCD to an optimal 3−GDPD of type gu, remove only +di (mod gu), where

i ∈ {0, 1, . . . , gu2d− 1} developments of the starter blocks to be deleted, and adjoin the

blocks to be added.

• (g, u) = (3, 8), d = 4:


{0, 2, 12}

{0, 6, 11}

{1, 12, 13}

{2, 3, 6}

{2, 22, 7}

{2, 13, 14}

{1, 3, 18}

{1, 19, 15}

{1, 11, 4}

{1, 23, 6}

{2, 8, 11}

Starter blocks to be deleted:{{0, 2, 12}, {1, 12, 13}, {2, 13, 14}}


Blocks to be added:

{2, 12, 13}

{5, 15, 16}

{8, 18, 19} {11, 21, 22}

• (g, u) = (3, 10), d = 3:


{1, 3, 27}

{2, 10, 28}

{2, 20, 25}

{0, 18, 13}

{2, 9, 17}

{0, 27, 11}

{0, 9, 26}

{2, 4, 3}

{1, 20, 15}

{0, 2, 19}

{1, 17, 16}

{1, 24, 9}

{1, 10, 4}

{2, 26, 23}

Starter blocks to be deleted:{{2, 9, 17}, {1, 17, 16}, {1, 24, 9}}

Blocks to be added:

{1, 9, 17}

{4, 12, 20}

{7, 15, 23}

{10, 18, 26}

{13, 21, 29}

7.2.2 No edges in excess (leave) graphs

We start with an easy observation: a GDD is both an optimal GDCD and an optimal

GDPD. The existence of 3−GDD is implied by Theorem 2.3.

Corollary 7.11. There exists a 3−GDD of type gu, which is an optimal 3−GDCD of

type gu and an optimal 3−GDPD of type gu, when u ≥ 3 and

• u ≡ 1, 3 (mod 6), or

• g ≡ 0 (mod 2) and u ≡ 0, 4 (mod 6), or


• g ≡ 0 (mod 3) and u ≡ 5 (mod 6), or

• g ≡ 0 (mod 6) and u ≡ 2 (mod 6).

7.2.3 Two edges in excess graphs

When either g ≡ 2, 4 (mod 6) and u ≡ 2 (mod 6), or g 6≡ 0 (mod 3) and u ≡ 5 (mod 6),

the excess graph of an optimal 3−GDCD of type gu has only two edges. Moreover, the

excess graph is one edge of multiplicity two, since g(u−1) ≡ 0 (mod 2). Hence, removing

any two out of three blocks which contain the same pair of elements, we get a 3−GDPD

with C(3, gu)− 2 = g2u(u−2)+46

− 2 = g2u(u−2)−86

= U(3, gu) blocks. Note that these are the

families of 3−GDPDs for which the Johnson bound is applicable. Therefore, we get an

optimal 3−GDPD of type gu, with four edges in the leave graph.

Theorem 7.12.

D(3, gu) =

⌊gu

3

⌊g(u− 1)

2

⌋⌋− 1,

when u ≥ 3 and

1. g ≡ 2, 4 (mod 6) and u ≡ 2 (mod 6), or

2. g 6≡ 0 (mod 3) and u ≡ 5 (mod 6).

7.2.4 One regular excess (leave) graphs

When either g ≡ 1 (mod 2) and u ≡ 0 (mod 6), or g ≡ 3 (mod 6) and u ≡ 2, 4 (mod 6),

both the excess graph and the leave graph of optimal 3 − GDCDs and 3 − GDPDs,

respectively, are one regular graphs. We consider constructions of objects in these

congruence classes next.

Lemma 7.13. Let g ≡ 1 (mod 2) and u ∈ {6, 12}. There exists an optimal 3−GDCD

of type gu which transforms into an optimal 3−GDPD of type gu.

Proof. When g = 1, a transferable 3 − GDCD of type 1g exists by Lemma 7.8. By

Theorem 2.14, there exists a 3 −HGDD of type (u, 1g) for any g ≡ 1 (mod 2), g ≥ 3.


Fill each hole with the transformable optimal 3−GDCD of type 1u. To get an optimal

3−GDPD, transform each copy of the optimal 3−GDCD of type 1u in the construction

to an optimal 3−GDPD of type 1u.

Next, we use Wilson’s Construction 6.4 to get a transformable 3−GDCD of type gu

for any u ≡ 0 (mod 6), u ≥ 18.

Theorem 7.14. Let g ≡ 1 (mod 2) and u ≡ 0 (mod 6), u ≥ 6. There exists an optimal

3−GDCD of type gu which transforms into an optimal 3−GDPD of type gu.

Proof. Lemma 7.13 gives a solution when u ≤ 12. Assume that u = 6l, l ≥ 3. Apply

Construction 6.4 using a 3−GDD of type 6l, which exists by Theorem 2.3 for all l ≥ 3,

and an optimal, essential 3 − GDCD of type g6, which exists by Lemma 7.13. To get

an optimal 3−GDPD of type gu, transform each 3−GDCD of type g6 to an optimal

3−GDPD of type g6.

We now consider the case g ≡ 3 (mod 6) and u ≡ 2, 4 (mod 6).

Lemma 7.15. Let u ≡ 2, 4 (mod 6), u ≥ 4. If there exists an optimal, essential

3 − GDCD of type 3u which transforms into a 3 − GDPD of type 3u with U(3, 3u)

blocks, then for any g ≡ 3 (mod 6), there exists an optimal 3−GDCD of type gu which

transforms into an optimal 3−GDPD of type gu with U(3, gu) blocks.

Proof. Let g = 6l + 3 = 3(2l + 1), l ≥ 1. By Theorem 2.14, there exists a 3 −HGDD

of type (u, 32l+1). Fill each hole with an optimal 3−GDCD of type 3u. Transforming

each copy of the ingredient 3 − GDCD of type 3u in the construction into an optimal

3−GDPD of type 3u, we get the desired optimal 3−GDPD.

Lemmas 7.9, 7.10, and 7.15 give the following corollaries.

Corollary 7.16. Let u ∈ {4, 8, 10} and g ≡ 3 (mod 6). There exists a optimal 3−GDCD

of type gu which transforms into an optimal 3−GDPD of type gu with U(3, gu) blocks.


Corollary 7.17. Let u ∈ {14, 16, 20} and g ≡ 3 (mod 6). If there exists an optimal,

essential 3−GDCD of type 3u which transforms into an optimal 3−GDPD of type 3u

with U(3, 3u) blocks, then there exists an optimal 3−GDCD of type gu which transforms

into an optimal 3−GDPD of type gu with U(3, gu) blocks.

We now use Wilson’s construction to get a transformable 3−GDCD for any u ≡ 2, 4

(mod 6), u ≥ 22.

Theorem 7.18. Let g ≡ 3 (mod 6) and u ≡ 2, 4 (mod 6), u ≥ 24. There exists an

optimal 3−GDCD of type gu which transforms into an optimal 3−GDPD of type gu

with U(3, gu) blocks, except possibly when u ∈ {14, 16, 20}.

Proof. Corollary 7.16 considers cases u ≤ 10. Let u = 6l+m, where l ≥ 3 and m ∈ {4, 8}.

Apply Construction 6.4 using a 3−GDD of type 6lm1, given in Theorem 2.3, an optimal,

essential 3−GDCD of type g6, constructed in Theorem 7.14, and an optimal, auxiliary

3−GDCD of type gm, which exists by Corollary 7.16.

To get an optimal 3−GDPD of type gu, transform each copy of the 3−GDCD of

type g6 and the copy of the 3−GDCD of type gm into an optimal 3−GDPD.

7.2.5 Almost one regular excess (leave) graphs

It remains to consider the cases g ≡ 1, 5 (mod 6) and u ≡ 2, 4 (mod 6) for which either

one of or both the excess and the leave graphs are irregular; they have almost all vertices

of degree one except one or two vertices of degree three.

Lemma 7.19. Let u ≥ 4, u ≡ 2, 4 (mod 6), and let m ∈ {1, 5}. If there exist an optimal,

essential 3−GDCD of type 3u and an optimal, auxiliary 3−GDCD of type mu, both of

which transform into optimal 3−GDPDs of types 3u and mu with U(3, 3u) and U(3,mu)

blocks, respectively, then there exists an optimal 3−GDCD of type gu which transforms

into an optimal 3 − GDPD of type gu with U(3, gu) blocks, for any g ≡ m (mod 6),

g 6∈ {7, 11}.


Proof. Let g = 6l + m = 3(2l) + m, l ≥ 2. By Theorems 2.5 and 2.14, there exist a

3−GDD of type 32lm1 and a 3−HGDD of type (u, 31). Hence, by Construction 2.16,

there exists a 3 − DGDD of type (u, 32lm1). Fill the holes with an optimal, essential

3−GDCD of type 3u, and an optimal, auxiliary 3−GDCD of type mu.

Since both ingredient 3−GDCDs transform into the optimal ingredient 3−GDPDs,

the product of the construction transforms into an optimal 3−GDPD of type gu.

Lemmas 7.8 and 7.19 and Theorem 7.18 imply the following corollary.

Corollary 7.20. Let u ∈ {10, 14, 16, 22}, and let m ∈ {1, 5}. If there exist an optimal

essential 3−GDCD of type 3u and an optimal, auxiliary 3−GDCD of type mu, both of

which transform into optimal 3− GDPDs of types 3u and mu, respectively, then there

exists an optimal 3−GDCD of type gu which transforms into an optimal 3−GDPD of

type gu with U(3, gu) blocks, where g ≡ m (mod 6), g 6∈ {7, 11}.

Next, we apply Wilson’s construction to obtain objects for a given g ≡ 1, 5 (mod 6)

and any u ≥ 22, u ≡ 2, 4 (mod 6).

Lemma 7.21. Let g ≡ 1, 5 (mod 6), and let m ∈ {4, 8}. If there exists an optimal,

auxiliary 3−GDCD of type gm which transforms into an optimal 3−GDPD of type gm

with U(3, gm) blocks, then there exists an optimal 3−GDCD of type gu which transforms

into an optimal 3 − GDPD of type gu with U(3, gu) blocks, where u ≡ m (mod 6),

u 6∈ {10, 14, 16, 20}.

Proof. Let u = 6l + m, l ≥ 3. Apply Construction 6.4 using a 3 − GDD of type 6lm1,

which is given in Theorem 2.5, an optimal, essential 3−GDCD of type g6, which is given

by Theorem 7.14, and an optimal, auxiliary 3−GDCD of type gm.

Since both ingredient 3−GDCDs transform into optimal, ingredient 3−GDPDs, the

product of the construction transforms into an optimal 3−GDPD of type gu as well.

Together Lemmas 7.19 and 7.21 imply the following theorem.


Theorem 7.22.

1. Let g0 ∈ {1, 5} and u0 ∈ {4, 8}. If there exists an optimal, essential 3 − GDCD

of type gu00 which transforms into an optimal 3−GDPD of type gu00 with U(3, gu00 )

blocks, then there exists an optimal 3−GDCD of type gu which transforms into an

optimal 3−GDPD of type gu with U(3, gu) blocks, where g ≡ g0 (mod 6), g ≥ 13,

and u ≡ u0 (mod 6), u ≥ 22.

2. Let g0 ∈ {7, 11}, and u0 ∈ {4, 8}. If there exists an optimal, essential 3−GDCD

of type gu00 which transforms into an optimal 3−GDPD of type gu00 with U(3, gu00 )

blocks, then there exists an optimal 3−GDCD of type gu0 which transforms into an

optimal 3−GDPD of type gu with U(3, gu) blocks, where u ≡ u0 (mod 6), u ≥ 22.

Finally, we apply Theorem 7.22 using the examples of transformable 3−GDCDs from

Lemma 7.8 and the results of Theorem 7.14.

Corollary 7.23. There exists an optimal 3−GDCD of type gu which transforms into

an optimal 3−GDPD of type gu with U(3, gu) blocks, when g ≡ 1 (mod 6) and u ≡ 2, 4

(mod 6), except possibly when g = 7 or u ∈ {10, 14, 16, 20}.

7.2.6 Optimal 3−GDPDs

To complete this study, we need to find some ingredients. In particular, we need optimal

3 − GDCDs of type gu which transform into optimal 3 − GDPDs, where g = 3 and

u ∈ {14, 16, 20}; g ∈ {5, 7, 11} and u ∈ {4, 8}; and g ∈ {1, 5} and u ∈ {10, 14, 16, 20}.

This is a challenging problem because of the requirement that the 3−GDCD transforms

to an optimal 3−GDPD. However, we are able to find the ingredient 3−GDPDs for

the constructions in Sections 7.2.4 and 7.2.5 directly. In this way, we construct optimal

3−GDPDs, but we do not have a method of transforming a packing to a covering and

vice versa for these families of objects.

In Appendix B, Theorems B.1 and B.2 give the examples of optimal ingredient


3−GDPDs. Lemma 7.15 and Theorem B.1 imply the following lemma.

Lemma 7.24. Let g ≡ 3 (mod 6) and u ∈ {14, 16, 20}. Then, there exists an optimal

3−GDPD of type gu with U(3, gu) blocks.

Theorem 7.22 and Theorems B.1 and B.2 imply the following lemma.

Lemma 7.25. Let g ≡ 5 (mod 6) and u ≡ 2, 4 (mod 6), u ≥ 4, and u 6∈ {10, 14, 16, 20}.

There exists an optimal 3−GDPD of type gu with U(3, gu) blocks.

Finally, Corollary 7.20, Theorem 2.20, and Theorem B.2 imply the lemma below.

Lemma 7.26. Let g ≡ 1, 5 (mod 6), g 6∈ {7, 11}, and let u ∈ {10, 14, 16, 20}. Then,

there exists an optimal 3−GDPD of type gu with U(3, gu) blocks.

7.2.7 Summary of results

We can summarize the results of Subsections 7.2.2-7.2.5 in the following theorem.

Theorem 7.27. There exists an optimal 3 − GDCD of type gu which transforms into

an optimal 3−GDPD of type gu with U(3, gu) blocks if u ≥ 3 and one of the following

holds:

• g ≥ 1 and u ≡ 0, 1, 3, 5 (mod 6),

• g ≡ 0, 2, 4 (mod 6) and u ≡ 2, 4 (mod 6),

• g ≡ 1 (mod 6) and u ≡ 2, 4 (mod 6), except possibly when g ∈ {7, 11} or u ∈

{10, 14, 16, 20},

• g ≡ 3 (mod 6) and u ≡ 2, 4 (mod 6), except possibly when u ∈ {10, 14, 16, 20}.

In Subsection 7.2.8, Table 7.4 summarizes the constructions of optimal 3−GDCDs

which transform into optimal 3−GDPDs. Table 7.3 gives the list of necessary ingredients

for the constructions which are not known to exist yet (cf. Corollaries 7.17, and 7.20, and

Theorem 7.22). Some ingredient 3−GDCDs have large parameters and we may require

new constructions for them.


g = 3, u ∈ {14, 16, 20} Corollary 7.17g = 5, u ∈ {4, 8} Theorem 7.22g ∈ {7, 11}, u ∈ {4, 8} Theorem 7.22g ∈ {1, 5}, u ∈ {10, 14, 16, 20} Corollary 7.20g ∈ {7, 11}, u ∈ {10, 14, 16, 20} Exceptions

Table 7.3: List of necessary ingredient, optimal 3−GDCDs of type gu which transforminto optimal 3−GDPDs of type gu, required for constructions in previous sections.

On the other hand, Lemmas 7.24-7.26, complete the problem of the existence of

optimal 3−GDPDs, though these 3−GDPDs do not come from optimal 3−GDCDs.

Together with Theorem 7.27, we get the following result which contains only eight possible

exceptions. We hope that the further study of transformable 3−GDCDs will yield a new

construction dependent on small ingredients which gives a solution to these exceptions

that require relatively big collections of blocks.

Theorem 7.28. Let g and u ≥ 3 be positive integers. Then,

D(3, gu) =⌊u3

⌊(u−1)

2

⌋⌋− δ, where δ =

1, g ≡ 2, 4 (mod 6) and u ≡ 2 (mod 6),

1, g 6≡ 0 (mod 3) and u ≡ 5 (mod 6),

0, otherwise,

except possibly when g ∈ {7, 11} and u ∈ {10, 14, 16, 20}.

7.2.8 Summary of 3-GDCD constructions

Table 7.4 summarizes the constructions of optimal 3 − GDCDs which transform into

optimal 3−GDPDs of the same type. The columns of the table are congruence classes g

(mod 6), and the rows of the table are the congruence classes u (mod 6). The column

g ≡ 0 (mod 6) is omitted, since these 3−GDCDs are well-known GDDs.

We give an example how to interpret the content of the table on the cell corresponding

to g ≡ 5 (mod 6) and u ≡ 2 (mod 6). The first line means that for a fixed value of

u ≡ 2 (mod 6), if there exists an optimal 3−GDCD of type gu0 for all g0 ∈ {3, 5} which


transforms into an optimal 3−GDPD of type gu0 , then there exists an optimal 3−GDCD

of type gu for any g ≥ 17, g ≡ 5 (mod 6) which transforms into an optimal 3−GDPD

of type gu (cf. Lemma 7.19.) The ingredients with g0 = 3 are essential. The second line

says that for a fixed value of g ≡ 5 (mod 6), if there exists an optimal 3 − GDCD of

type gu0 for all u0 ∈ {6, 8}, which transforms into an optimal 3 − GDPD of type gu0 ,

then there exists an optimal 3−GDCD of type gu for any u ≥ 24, u ≡ 2 (mod 6), which

transforms into an optimal 3−GDPD of type gu (cf. Lemma 7.21.) The ingredients with

u0 = 6 are essential. Finally, the last line denotes that to apply these two constructions,

we need to determine the existence of an optimal, transformable 3−GDCD of type 58.


Tab

le7.

4:Sum

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const

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const

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(mod

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3g

=1∗⇒

g≥

5u

=6∗⇒

u≥

18u

=6∗⇒

u≥

18u

=6∗⇒

u≥

181 2

g=

1,3∗⇒

g≥

13D

elet

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g=

3∗⇒

g≥

9D

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13u

=6∗,8⇒

u≥

26w

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u=

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u≥

26w

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u≥

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ssM

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)=

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g=

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g≥

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g=

3∗⇒

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g=

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22w

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toex

cess

Chapter 8

Conclusion

We have studied a family of combinatorial objects which can be represented in three

equivalent forms: as an array, as a triple of sets, or as a graph covering problem. In the

array representation, we call them covering arrays with row limits, CARLs. When strength

t = 2, in the set representation, they are called group divisible covering designs, GDCDs,

and in the graph representation, they are equivalent to a version of the graph covering

problem. Here, we talk about CARLs while keeping in mind that they are equivalent to

two other problems. In each representation, CARLs are a natural generalization of well

studied families of objects or problems such as: covering arrays, group divisible designs,

and graph decompositions. As such, CARLs inherit important characteristics of each of

these objects, and different representations bring out different aspects of the study of

CARLs.

The array representation is convenient for treating CARLs as test suites for interaction

testing in which there is a limit to the number of components tested at a time. This is the

main application of covering arrays. Some properties and constructions of covering arrays

can be generalized for CARLs, such as logarithmic growth and the product construction.

We have observed that some ‘fill in one cell at a time’ techniques applicable to covering

arrays, are not adaptable as such to CARLs.

145

Chapter 8. Conclusion 146

On the other hand, when the row limit is a constant, and the number of columns

and their alphabet size varies, it is more advantageous to use the set representation of

CARLs, i.e. GDCDs. In this case, a row of such a CARL would have many more empty

cells than non-empty ones. It is more convenient to record the non-empty entries in a

form of a block of the constant size.

8.1 Summary of results

The dissertation consists of three studies. The first one considers the question of the size

of a CARL by studying the lower and upper bounds. We also introduce three construction

methods for CARLs. In the second study our objective has been to construct optimal

CARLs with row limit four. More precisely, we have constructed 4−GDCDs. Finally,

the third study considers PARLs and GDPDs. Here we derive two upper bounds on the

size of PARLs and we consider a transformation of optimal 3 − GDCDs into optimal

3−GDPDs.

We have shown that if the row limit w(k) is a function of the number of columns k,

for different functions, CARLs have different growth. In particular, if w(k) = Θ(k), then

CARLs have a logarithmic growth (cf. Corollaries 4.1 and 4.5), which is a characteristic

of covering arrays. In this case, we do not have any other lower bound other than the

size of a covering array with the same parameters (see Chapter 4). On the other hand,

if w(k) = o(k), then the rate of growth of CARLs is at least as much as the growth

of the Schonheim lower bound, Theorem 4.2, which is much greater than logarithmic.

If we restrict the row limit to be a constant function, the rate of growth is Θ(kt), and

the Schonheim bound is asymptotically optimal (cf. Theorem 4.7). The same holds for

PARLs with constant row limit, as shown in Theorem 7.2. The Johnson bound adjusts

the Schonheim bound by a constant for a number of families of CARLs and PARLs (cf.

Section 7.1.3).


We have derived two probabilistic upper bounds on the size of any CARL in Chapter 4.

In general, the binomial upper bound UB1 in Theorem 4.4 is better than uniform

distribution upper bound UB0 given in Theorem 4.3. UB1 is also an improvement for the

upper bounds on the size of a covering array for a spectrum of values of k (cf. Section 4.3).

With the constraint w(k) ln(w(k)) = o(k), Theorem 4.6 gives a constructive upper bound

which is a slight improvement of UB1.

In Section 5.1, we give a greedy algorithm for the construction of CARLs whose size

is at most equal to UB0. This algorithm runs in polynomial time when w(k) is a constant

function. However, depending on w(k), the algorithm may run in exponential time, which

in the best case is useful; but in the worst case, the constructed object may exceed the

upper bound UB1. The proof of Theorem 4.6 gives a probabilistic construction of CARLs

when w(k) is such that w(k) lnw(k) = o(k). We also have two combinatorial constructions

of CARLs: Product Construction 5.2 for strength t = 2, and Wilson’s Construction 5.6.

We have applied these constructions to obtain families of CARLs with a constant ratio

kw(k)

(See Sections 5.2 and 5.3).

Furthermore, we have used Wilson’s construction and the structure of the excess graph

to find optimal 4−GDCDs and 3−GDCDs which transform into optimal 3−GDPDs.

We have proved that when w = 4, there exists a constant δ > 0, such that the size of an

optimal 4−GDCD differs from the Schonheim lower bound by at most δ blocks, with

some possible exceptions (see Theorem 6.2). Also we have constructed many families of

optimal 4−GDCDs (cf. Theorem 6.59) and two families of close to optimal 4−GDCDs

(cf. Theorem 6.60). Given constructions have the potential of finding an optimal solution

to all 4 − GDCDs with possible exceptions given in Theorem 6.60. However, there

is a sizable list of necessary ingredients whose optimal size is still undetermined (see

Appendix A). Section 6.6 gives an alternative method of construction of some of the

families of 4 − GDCDs which requires more essential ingredients, but overall fewer

necessary ingredients in total.


Finally, we have began a study of packing arrays with row limit, PARLs. The results

on CARLs with constant row limit imply analogous results for PARLs (cf. Theorem 7.2).

The main idea behind the proof of Theorem 7.2, which proves that the Schonheim upper

bound is asymptotically equal to the optimal size of PARLs with constant row limit,

suggest that optimal GDCDs can be easily transformed into optimal GDPDs. We

explored this idea in Section 7.2 where we constructed all but two classes of transformable

3−GDCDs into optimal 3−GDPDs (see Theorem 7.27). Furthermore, Theorem 7.28

determines the optimal size of 3−GDPDs of type gu for positive integers g and u ≥ 3

with only eight possible exceptions, some of which might not be transformable. For the

study of optimal 3 − GDPDs, we have had to strengthen the Schonheim bound for a

number of congruence cases by the Johnson bound ( cf. Theorem 7.3). The Johnson

bound also applies to CARLs (see Theorem 7.5).

8.2 Future work and open questions

Though we proved many interesting results, in the course of our research we discovered

even more intriguing open questions and possible further areas of study.

The extensive study of bounds on the size of covering arrays and the fact that in

general, the optimal size of a covering array is unknown imply that the question of the

optimal size of CARLs when w(k) = Θ(k) is a difficult problem. Even though we have an

upper bound which improves an important bound for covering arrays for a spectrum of

values of k, we do not know if it is optimal for any family of CARLs. Theorems 5.8 and 5.5

give families of CARLs for which the size is considerably smaller than predicted by UB1

(see Theorem 4.4) implying that the bound UB1 can probably be further strengthened.

This is affirmed by the examples with small values of the parameters v and t, in which

the bound UB0 is smaller than UB1. We also do not know if the improvement of UB1

in Theorem 4.6 is optimal for any family of CARLs, or if this bound too can be further


strengthened.

Moreover, there is an open question of the lower bounds for CARLs when w(k) =

ω(

k(log k)1/t

). In this case, the only lower bound we have is the size of a covering array

with the same parameters. Otherwise, the Schonheim bound seems to be a good lower

bound. An interesting question is whether the Schonheim lower bound is optimal for any

positive integer valued function w = w(k) which is not a constant function. We know

that w has to be such that w(k) = o(k), but we do not know how small it has to be.

In Sections 5.3 and 5.2, we have constructed several families of CARLs. We do not know

if any of them are optimal. In particular, the CARLs constructed in Theorems 5.7 and 5.9

are close to the Schonheim lower bound. Given that in these examples the parameters of

CARLs are functions of prime powers, and the CARLs are constructed from orthogonal

arrays, one might be able to further improve these constructions and obtain CARLs

which meet the Schonheim lower bound. In this case there would be a family of objects

which have a non-empty excess graph, for which w(k) = Θ(k) and v = v(k), and which

meet the Schonheim lower bound.

Originally, Wilson’s construction is used for group divisible designs. Its generalizations

are applicable for construction of CARLs of any strength (cf. Construction 5.6), and

it is used to build optimal families of CARLs with fixed row limits (cf. Chapter 6 and

Section 7.2). There are probably other constructions, in particular constructions of

t-designs, that can be generalized and used in the context of CARLs.

The Deterministic Density Algorithm, DDA, is a polynomial time algorithm for the

construction of covering arrays [6,13]. However, it is based on ‘filling in one cell at a time,’

a method which cannot be generalized as is when we have a row limit (see Section 5.1).

We have a greedy, possibly exponential time algorithm for construction of CARLs, which

are also large in size. Another direction of further study is to find a polynomial time

algorithm for construction of CARLs which do not exceed the upper bound UB1 given in

Theorem 4.4.


We have began the study of PARLs and related objects. The results and construction

methods which we have for PARLs are directly implied by the results and constructions

of CARLs for which the row limit is a constant. We know little of how the size of a

PARL changes for different functions w(k). We have solved the problem of optimal

3 − GDPDs, but we are still missing some optimal ingredients in the construction of

optimal 3−GDCDs which transform into optimal 3−GDPDs. GDCDs and GDPDs

with a regular excess or leave graph are easier to find by a computer search even when

they have large parameters. We have seen that in some cases, an optimal GDCD has a

regular graph where an optimal GDPD does not, and vice versa, hence it can be easier

to find one of them. Then we just need to transform one into another. When w(k) is not

a constant, the relationship between optimal CARLs and PARLs might not be as simple.

However, one or another may be easier to construct. Also, the question of constructive

methods, both combinatorial and algorithmic, for PARLs is wide open.

Finally, packing arrays are related to maximum-distance-separable (MDS) codes [49],

whereas covering arrays are related to surjective codes [10]. It would be interesting

to explore relationships between CARLs, PARLs, and codes, in terms of bounds and

constructions.

Appendices

151

Appendix A

List of missing ingredients

Here we list the ingredient 4−GDCDs for constructions in Section 6.4 and in Section 6.5

whose optimal size is still undetermined. The ingredients are listed by whether they are

essential or auxiliary objects in the constructions, and by the excess graph they would

have if they met the Schonheim lower bound (cf. Theorem 6.1).

In the following tables, we use “E” to denote an essential ingredient, and “A” to

denote an auxiliary ingredient in the construction.

152

Appendix A. List of missing ingredients 153

Ingredients in constructions of two-dimensional families of 4−GDCDs:

Regular excess graph

E (g, u) = (8, 23) Thm. 6.52 and Thm. 6.53

E (g, u) = (17, 12) Thm. 6.47

A g ∈ {20, 44}, u ∈ {11, 17, 23} Cor. 6.33, Thm. 6.50

A (g, u) = (44, 5) Cor. 6.33, Thm. 6.50

Irregular excess graph

A g ∈ {7, 13, 19}, u = 5 Thm. 6.27

A g ∈ {7, 10, 13, 19, 22}, u = 9 Thm. 6.46

A g ∈ {14, 26, 38}, u ∈ {5, 11, 17, 23} Cor. 6.33, Thm. 6.50

A g = 2, u ∈ {17, 23} Cor. 6.33

A g ∈ {5, 11, 17, 23, 29, 35, 41, 47}, u ∈ {5, 11, 23, 17} Cor. 6.41, Thm. 6.53

A g ∈ {5, 11, 17}, u ∈ {6, 9} Thm. 6.47

Ingredients for constructions of families of 4−GDCDs with a fixed value of u:


A g = 4, u ∈ {21, 27} Cor. 6.28

E g = 8, u ∈ {29, 35, 41, 47} Cor. 6.33

A g = 20, u ∈ {29, 35, 41, 47} Cor. 6.33

A g = 5, u ∈ {27, 36, 39, 51} Cor. 6.37


A (g, u) = (5, 14) Thm. 6.27

A g ∈ {7, 10}, u ∈ {15, 21, 27} Cor. 6.28

A g ∈ {2, 14}, u ∈ {29, 35, 41, 47} Cor. 6.33

A g ∈ {5, 11, 17, 23}, u ∈ {29, 35, 41, 47} Cor. 6.41

Appendix A. List of missing ingredients 154

Ingredients for constructions of families of 4−GDCDs with a fixed value of g:


A (g, u) = (17, 26) Thm. 6.52


A g ∈ {5, 7}, u = 7 Cor. 6.43

Individual cases with fixed values of g and u omitted by the constructions:

g u Reference

13, 19 14 Thm. 6.27

13,19,22 15, 21,27 Cor. 6.28

26,38,44 29,35,41,47 Cor. 6.33

29,35,41,47 29,35,41,47 Cor. 6.41

5,7 10,19 Cor. 6.43

11,17 27,36,39,51 Cor. 6.37

11,17 32,38,44,50 Thm. 6.52

Appendix B

Some optimal 3−GDPDs

We give solutions to some optimal 3−GDPDs here. For definition of the packing number

D(k, gu), see Definition 3.8.

Theorem B.1. Let g = 3 and u ∈ {14, 16, 20}, or (g, u) = (7, 8). Then,

D(3, gu) =

⌊gu

3

⌊g(u− 1)

2

⌋⌋.

Proof. Let V = Zgu be the set of elements. Let the group Gi = {x ∈ Zgu : x ≡ i

(mod u)}, i ∈ Zu. In each case, we give an integer d. Develop the blocks +d (mod gu).

g = 3, u = 14, d = 3:

{0, 7, 6}{0, 37, 8}{0, 24, 40}{0, 4, 41}{2, 22, 33}

{1, 9, 41}{1, 34, 11}{1, 16, 5}{0, 13, 29}{0, 27, 30}

{0, 25, 26}{0, 17, 23}{0, 20, 35}{0, 10, 2}{1, 13, 37}

{1, 24, 4}{2, 41, 11}{1, 8, 26}{0, 5, 9}

g = 3, u = 16, d = 3:

{1, 47, 42}{2, 23, 21}{1, 46, 5}{0, 9, 19}{1, 41, 2}{2, 13, 24}

{0, 22, 44}{2, 37, 31}{2, 44, 14}{1, 18, 19}{0, 11, 12}{0, 28, 40}

{0, 23, 8}{1, 3, 24}{0, 20, 34}{1, 11, 28}{0, 42, 35}{0, 30, 45}

{2, 36, 5}{1, 44, 16}{1, 36, 26}{0, 43, 4}

155

Appendix B. Some optimal 3−GDPDs 156

g = 3, u = 20, d = 3:

{10, 0, 7}{8, 39, 49}{6, 50, 12}{0, 59, 4}{0, 16, 43}{2, 9, 51}{2, 34, 36}{1, 42, 54}{10, 47, 33}{7, 14, 50}{7, 8, 19}{0, 15, 32}{1, 7, 15}{0, 24, 25}{11, 45, 24}{11, 32, 23}{5, 24, 30}{10, 21, 36}{8, 2, 29}{4, 49, 57}{10, 46, 25}{10, 11, 17}{8, 38, 32}{8, 55, 22}{7, 18, 42}{11, 15, 41}{7, 59, 38}{11, 43, 7}

{4, 42, 56}{3, 12, 9}{6, 38, 43}{8, 18, 33}{8, 5, 46}{2, 5, 17}{6, 2, 49}{0, 55, 53}{4, 35, 46}{8, 24, 35}{9, 45, 36}{6, 14, 11}{9, 11, 21}{5, 1, 6}{6, 59, 35}{6, 58, 45}{7, 9, 41}{1, 11, 28}{11, 36, 54}{8, 37, 21}{9, 38, 52}{1, 3, 53}{7, 36, 32}{1, 56, 8}{2, 10, 14}{6, 53, 24}{10, 39, 6}{1, 34, 59}

{2, 44, 53}{9, 19, 6}{4, 55, 36}{3, 26, 45}{1, 23, 17}{8, 11, 30}{4, 54, 20}{7, 39, 46}{9, 20, 37}{5, 27, 26}{5, 22, 4}{8, 0, 50}{2, 57, 43}{1, 19, 49}{11, 2, 52}{2, 48, 25}{5, 54, 55}{6, 36, 3}{3, 39, 16}{7, 4, 58}{9, 13, 39}{1, 50, 35}{4, 19, 15}{1, 55, 10}{10, 5, 20}{9, 15, 17}{4, 12, 25}{11, 55, 39}

{8, 9, 40}{4, 13, 10}{2, 1, 37}{5, 16, 52}{6, 33, 34}{9, 18, 34}{10, 44, 40}{4, 39, 26}{3, 47, 24}{6, 52, 51}{3, 8, 15}{1, 29, 43}{0, 3, 22}{5, 51, 42}{4, 45, 2}{9, 32, 47}{11, 25, 3}{5, 47, 29}{5, 0, 48}{5, 40, 28}{9, 43, 4}{10, 41, 8}{9, 53, 31}{2, 28, 35}{3, 44, 42}{3, 46, 58}{4, 30, 1}{0, 46, 2}

g = 7, u = 8, d = 1:

{0, 21, 27}{0, 11, 36}

{0, 30, 18}{0, 55, 14}

{0, 17, 22}{0, 19, 52}

{0, 53, 10}{0, 47, 49}


Theorem B.2. Let g ∈ {5, 7, 11} and u = 4, or g ∈ {5, 11} and u = 8, or g = 5 and

u ∈ {10, 14, 16, 20}. Then,

D(3, gu) =

⌊gu

3

⌊g(u− 1)

2

⌋⌋.

Proof. Let V = Zgu be the set of elements. Let the groups be Gi = {x ∈ Zgu : x ≡ i

(mod u)}, i ∈ Zu. Below, we list the collection of blocks of an optimal 3−GDPD of type

gu for each case.

g = 5, u = 4:

{4, 5, 15}{11, 8, 14}{8, 5, 10}{16, 9, 7}{13, 11, 16}{13, 7, 12}{6, 15, 9}{2, 19, 12}{4, 2, 13}{11, 2, 1}{9, 4, 10}{0, 3, 18}

{7, 5, 14}{14, 4, 3}{19, 1, 4}{5, 18, 11}{18, 9, 12}{16, 5, 3}{0, 17, 15}{9, 0, 11}{2, 16, 17}{17, 18, 19}{1, 16, 18}{15, 8, 2}

{8, 18, 7}{6, 13, 3}{7, 1, 10}{2, 5, 0}{10, 15, 12}{16, 15, 14}{16, 19, 10}{19, 13, 8}{13, 10, 0}{18, 13, 15}{1, 14, 0}{17, 11, 10}

{14, 9, 19}{17, 7, 4}{4, 11, 6}{1, 6, 8}{2, 3, 9}{6, 5, 19}{3, 17, 8}{12, 17, 6}{1, 12, 3}{7, 6, 0}


g = 7, u = 4:

{14, 9, 0}{25, 24, 6}{15, 17, 12}{1, 4, 26}{1, 12, 23}{14, 23, 4}{27, 4, 6}{19, 17, 18}{22, 16, 1}{17, 23, 16}{15, 14, 5}{24, 27, 26}{15, 26, 25}{16, 13, 15}{1, 3, 18}{2, 27, 13}{3, 0, 13}{24, 2, 19}{5, 19, 8}{8, 22, 3}{16, 9, 26}{24, 10, 15}{0, 1, 10}{22, 9, 7}

{23, 24, 9}{24, 5, 3}{22, 12, 19}{6, 23, 0}{5, 18, 4}{7, 10, 16}{1, 27, 14}{2, 15, 1}{6, 19, 21}{7, 26, 5}{6, 15, 9}{7, 20, 6}{18, 0, 21}{26, 21, 23}{16, 14, 19}{26, 17, 8}{13, 18, 24}{11, 8, 14}{9, 20, 3}{1, 20, 19}{12, 2, 11}{5, 10, 23}{23, 22, 25}{4, 22, 17}

{8, 15, 21}{14, 7, 13}{21, 11, 10}{27, 12, 9}{24, 7, 1}{0, 27, 17}{20, 18, 15}{19, 26, 0}{8, 25, 7}{11, 22, 13}{7, 12, 18}{4, 10, 13}{3, 10, 25}{9, 11, 4}{16, 18, 27}{12, 26, 13}{10, 9, 19}{2, 4, 3}{12, 21, 3}{11, 16, 25}{11, 20, 26}{11, 0, 5}{12, 5, 6}{2, 23, 8}

{3, 14, 17}{23, 13, 20}{20, 21, 22}{8, 9, 18}{5, 27, 22}{12, 14, 25}{4, 7, 21}{6, 3, 16}{14, 21, 24}{2, 16, 21}{6, 13, 8}{10, 27, 8}{11, 17, 24}{2, 5, 20}{11, 1, 6}{17, 10, 20}{25, 27, 20}{2, 17, 7}{25, 2, 0}{22, 15, 0}{19, 4, 25}


g = 11, u = 4:

{8, 22, 15}{21, 2, 15}{25, 14, 20}{42, 13, 43}{26, 9, 36}{12, 27, 2}{28, 10, 17}{20, 18, 9}{6, 19, 16}{13, 28, 15}{3, 16, 1}{36, 42, 29}{30, 23, 1}{17, 15, 24}{12, 39, 26}{25, 4, 19}{14, 24, 7}{7, 22, 25}{24, 18, 29}{38, 3, 17}{19, 40, 1}{9, 15, 12}{30, 0, 29}{19, 41, 34}{34, 0, 1}{3, 21, 20}{35, 42, 32}{5, 26, 28}{25, 2, 24}{26, 20, 15}{16, 2, 33}{0, 43, 9}{33, 26, 0}{17, 12, 42}{34, 15, 16}{36, 35, 2}{41, 42, 15}{10, 15, 29}{5, 34, 7}{11, 21, 40}{37, 34, 23}{9, 7, 2}{42, 1, 39}{31, 10, 8}{12, 6, 5}

{8, 41, 26}{2, 23, 5}{37, 7, 42}{6, 15, 32}{38, 33, 15}{1, 22, 32}{16, 21, 14}{6, 37, 24}{17, 7, 26}{9, 30, 31}{21, 27, 28}{2, 29, 20}{1, 2, 8}{38, 23, 29}{32, 33, 23}{33, 20, 11}{18, 31, 5}{33, 30, 39}{32, 29, 31}{2, 40, 37}{43, 38, 24}{32, 17, 34}{38, 0, 41}{4, 23, 42}{29, 3, 34}{23, 17, 36}{22, 4, 33}{34, 27, 24}{14, 0, 27}{9, 32, 14}{4, 35, 30}{0, 7, 21}{13, 22, 40}{35, 28, 34}{36, 21, 10}{28, 9, 38}{28, 31, 25}{43, 6, 21}{20, 37, 19}{16, 30, 37}{43, 41, 4}{13, 26, 16}{8, 17, 27}{3, 10, 41}{37, 26, 11}

{21, 31, 26}{27, 22, 9}{26, 25, 23}{23, 12, 13}{23, 8, 14}{10, 25, 39}{21, 4, 34}{37, 27, 32}{43, 33, 12}{39, 40, 29}{4, 26, 29}{39, 8, 37}{1, 27, 26}{4, 39, 9}{36, 25, 15}{3, 40, 42}{11, 28, 42}{39, 34, 20}{31, 4, 38}{31, 2, 0}{10, 24, 33}{29, 12, 19}{38, 1, 11}{35, 5, 14}{30, 17, 40}{18, 25, 3}{43, 28, 14}{17, 35, 16}{7, 16, 41}{19, 30, 36}{14, 36, 3}{29, 6, 28}{10, 5, 27}{2, 39, 41}{33, 35, 18}{14, 40, 41}{13, 38, 27}{4, 14, 13}{12, 11, 10}{3, 28, 37}{15, 37, 0}{3, 9, 24}{18, 4, 27}{4, 10, 37}{10, 43, 16}

{35, 24, 26}{43, 34, 36}{42, 19, 21}{38, 16, 25}{18, 40, 15}{8, 42, 9}{38, 39, 36}{0, 19, 22}{6, 35, 41}{24, 5, 39}{13, 30, 20}{35, 8, 13}{6, 1, 36}{6, 7, 4}{1, 14, 12}{21, 12, 18}{12, 22, 41}{6, 31, 17}{25, 12, 35}{20, 22, 17}{11, 14, 29}{23, 18, 28}{4, 5, 11}{38, 35, 21}{36, 37, 18}{31, 34, 12}{40, 6, 27}{19, 17, 18}{8, 11, 18}{5, 36, 22}{33, 27, 42}{38, 19, 8}{4, 1, 15}{43, 18, 1}{2, 3, 4}{20, 1, 35}{21, 23, 22}{29, 27, 16}{10, 23, 20}{43, 22, 29}{31, 14, 37}{20, 42, 31}{6, 8, 25}{28, 22, 39}{19, 5, 32}


{38, 5, 40}{7, 20, 38}{36, 41, 11}{6, 13, 39}{13, 10, 32}{27, 20, 41}{33, 8, 34}{39, 14, 17}{31, 1, 24}{40, 33, 31}{42, 5, 16}{7, 18, 13}{39, 16, 18}{25, 34, 40}

{32, 2, 11}{7, 32, 30}{25, 11, 30}{2, 43, 17}{32, 41, 18}{23, 16, 9}{28, 7, 1}{32, 39, 21}{32, 25, 43}{23, 0, 6}{6, 3, 33}{33, 14, 19}{24, 11, 22}{6, 9, 11}

{26, 43, 40}{2, 28, 19}{10, 35, 0}{33, 7, 36}{30, 3, 12}{5, 3, 8}{13, 34, 11}{22, 37, 35}{0, 42, 25}{32, 26, 3}{22, 16, 31}{38, 37, 12}{30, 21, 24}{9, 10, 19}

{24, 13, 19}{0, 3, 13}{41, 24, 23}{8, 29, 7}{15, 5, 30}{41, 30, 28}{43, 5, 20}{7, 10, 40}{36, 13, 31}{11, 17, 0}{40, 9, 35}{43, 30, 8}


g = 5, u = 8:

{4, 5, 19}{36, 13, 34}{37, 4, 26}{27, 29, 33}{20, 31, 27}{16, 33, 39}{6, 39, 20}{14, 13, 24}{8, 26, 7}{34, 35, 23}{29, 26, 35}{1, 0, 26}{31, 34, 38}{23, 30, 9}{10, 1, 24}{12, 39, 29}{24, 28, 33}{19, 34, 39}{35, 22, 33}{7, 34, 12}{20, 3, 7}{10, 17, 30}{21, 14, 0}{12, 19, 17}{6, 3, 1}{17, 18, 23}{39, 9, 35}{9, 34, 21}{12, 3, 2}{31, 32, 13}{36, 38, 39}{6, 36, 9}{35, 30, 32}{3, 30, 37}{15, 0, 37}{8, 9, 11}{28, 31, 18}{16, 2, 31}{12, 24, 35}{17, 36, 8}{4, 24, 11}{25, 20, 13}{34, 33, 0}{6, 17, 15}{17, 21, 3}

{28, 34, 6}{1, 4, 29}{8, 37, 19}{17, 20, 5}{37, 10, 28}{32, 1, 12}{37, 38, 9}{28, 3, 39}{11, 26, 30}{1, 14, 31}{15, 25, 21}{31, 10, 6}{1, 23, 16}{24, 37, 2}{7, 2, 33}{15, 13, 18}{24, 31, 21}{28, 38, 25}{31, 5, 3}{20, 32, 21}{13, 38, 16}{0, 19, 18}{1, 28, 7}{32, 38, 5}{4, 15, 16}{23, 26, 21}{22, 39, 2}{4, 32, 27}{8, 22, 1}{15, 30, 2}{0, 5, 10}{33, 3, 4}{29, 16, 9}{4, 18, 8}{18, 6, 37}{20, 29, 23}{17, 29, 22}{25, 29, 18}{35, 25, 4}{23, 12, 5}{18, 38, 33}{30, 29, 0}{30, 7, 36}{7, 9, 18}{7, 14, 16}

{37, 1, 34}{35, 0, 7}{22, 36, 5}{21, 10, 11}{16, 11, 34}{14, 28, 23}{26, 15, 38}{26, 33, 32}{7, 38, 11}{4, 10, 14}{12, 38, 21}{17, 35, 14}{2, 4, 38}{34, 20, 30}{34, 29, 3}{3, 9, 32}{23, 22, 24}{1, 18, 39}{12, 27, 14}{39, 17, 4}{27, 1, 15}{0, 22, 13}{6, 11, 29}{22, 10, 16}{30, 19, 33}{10, 35, 20}{10, 3, 38}{21, 2, 1}{1, 38, 35}{0, 20, 2}{35, 15, 5}{17, 34, 32}{26, 36, 25}{35, 8, 31}{26, 28, 27}{19, 16, 26}{34, 5, 8}{5, 6, 26}{38, 20, 19}{20, 24, 18}{13, 2, 9}{14, 36, 3}{14, 20, 8}{31, 4, 9}{24, 39, 30}

{13, 19, 6}{7, 10, 19}{2, 29, 14}{6, 2, 32}{21, 30, 4}{18, 27, 21}{5, 33, 11}{23, 25, 32}{25, 0, 3}{28, 30, 8}{16, 21, 36}{12, 30, 25}{11, 18, 36}{17, 16, 28}{32, 37, 39}{14, 26, 39}{6, 35, 16}{37, 20, 16}{15, 33, 12}{29, 19, 31}{11, 1, 20}{36, 29, 24}{26, 24, 3}{17, 0, 31}{28, 13, 35}{28, 29, 15}{9, 0, 28}{28, 2, 5}{34, 15, 24}{8, 3, 15}{19, 1, 36}{31, 11, 12}{11, 39, 0}{10, 27, 13}{30, 1, 13}{33, 10, 23}{22, 4, 34}{2, 35, 36}{32, 36, 10}{31, 33, 36}{2, 23, 19}{24, 27, 6}{4, 7, 13}{39, 25, 10}{7, 24, 5}


{27, 39, 5}{3, 13, 23}{8, 21, 39}{28, 22, 32}{17, 37, 7}{25, 16, 5}{22, 31, 26}{19, 22, 25}{37, 36, 27}{38, 8, 29}{14, 34, 25}{2, 27, 17}

{24, 19, 9}{26, 13, 12}{5, 30, 18}{27, 22, 9}{14, 9, 5}{12, 18, 16}{25, 37, 31}{17, 24, 38}{12, 10, 8}{0, 38, 27}{8, 25, 6}{37, 23, 11}

{19, 21, 28}{27, 16, 30}{25, 7, 27}{11, 17, 13}{20, 9, 26}{6, 21, 33}{32, 18, 14}{29, 32, 7}{33, 13, 8}{15, 22, 20}{6, 23, 4}{21, 7, 22}

{3, 18, 22}{22, 37, 12}{23, 27, 8}{32, 15, 11}{14, 19, 15}{37, 33, 14}{6, 12, 0}{10, 15, 9}{0, 23, 36}{2, 25, 11}


g = 11, u = 8:

{1, 58, 3}{44, 78, 21}{38, 66, 4}{11, 50, 31}{49, 36, 46}{34, 39, 25}{43, 69, 66}{32, 29, 19}{17, 52, 71}{27, 6, 32}{72, 67, 70}{83, 73, 58}{54, 2, 72}{62, 35, 36}{54, 68, 21}{39, 77, 74}{33, 6, 26}{5, 36, 57}{85, 60, 42}{87, 21, 72}{77, 52, 83}{11, 71, 34}{48, 9, 54}{33, 19, 84}{34, 3, 24}{7, 26, 14}{51, 73, 79}{82, 47, 28}{5, 3, 81}{74, 57, 23}{25, 74, 80}{56, 20, 26}{50, 75, 77}{59, 9, 82}{54, 71, 43}{81, 71, 82}{59, 22, 74}{59, 73, 45}{78, 27, 29}{18, 44, 65}{49, 3, 79}{1, 38, 18}{39, 64, 83}{73, 10, 87}{8, 17, 59}

{40, 18, 70}{61, 27, 38}{20, 75, 32}{39, 62, 3}{60, 38, 53}{31, 58, 19}{32, 10, 11}{3, 73, 37}{67, 30, 20}{3, 66, 15}{4, 11, 40}{46, 87, 24}{63, 27, 17}{44, 54, 15}{64, 21, 23}{39, 17, 40}{79, 1, 85}{76, 25, 64}{51, 36, 66}{25, 40, 50}{83, 61, 31}{26, 24, 77}{49, 52, 85}{8, 53, 42}{75, 45, 10}{67, 68, 1}{25, 36, 77}{49, 70, 21}{57, 28, 3}{51, 37, 20}{17, 76, 15}{19, 30, 0}{31, 12, 51}{9, 0, 4}{50, 62, 27}{7, 45, 41}{47, 67, 33}{13, 60, 33}{6, 18, 67}{70, 10, 63}{65, 60, 23}{10, 17, 44}{29, 52, 73}{27, 0, 14}{28, 53, 7}

{12, 2, 65}{87, 56, 35}{58, 61, 7}{84, 41, 21}{68, 66, 64}{6, 42, 55}{80, 86, 31}{45, 6, 81}{17, 75, 72}{44, 63, 42}{58, 59, 41}{17, 43, 23}{33, 44, 31}{4, 7, 78}{59, 78, 53}{68, 46, 18}{22, 21, 8}{20, 38, 33}{21, 18, 43}{7, 5, 67}{7, 64, 2}{34, 36, 61}{35, 79, 12}{83, 6, 12}{73, 86, 40}{87, 42, 37}{4, 81, 42}{42, 23, 41}{43, 16, 36}{31, 2, 9}{71, 46, 25}{66, 16, 63}{77, 71, 73}{54, 58, 45}{0, 54, 82}{12, 73, 15}{54, 8, 13}{87, 19, 66}{61, 73, 74}{74, 85, 31}{77, 22, 76}{10, 61, 80}{67, 74, 84}{77, 12, 62}{32, 39, 84}

{45, 16, 17}{68, 82, 23}{72, 63, 85}{24, 29, 20}{11, 65, 5}{63, 26, 8}{2, 83, 71}{77, 19, 78}{83, 17, 70}{60, 26, 17}{38, 35, 50}{25, 29, 72}{67, 56, 31}{34, 68, 80}{81, 13, 78}{70, 64, 47}{29, 22, 67}{31, 26, 3}{59, 80, 63}{26, 65, 13}{76, 42, 14}{18, 33, 23}{8, 60, 83}{69, 1, 54}{62, 25, 53}{60, 48, 59}{41, 60, 11}{1, 15, 37}{20, 57, 69}{8, 66, 44}{71, 80, 12}{62, 4, 5}{77, 57, 55}{77, 15, 67}{22, 17, 0}{2, 17, 36}{66, 76, 27}{38, 10, 7}{40, 78, 79}{24, 41, 14}{43, 20, 63}{68, 7, 8}{14, 87, 67}{69, 30, 47}{23, 56, 53}


{41, 2, 51}{48, 34, 78}{73, 68, 11}{50, 85, 0}{69, 27, 8}{42, 40, 20}{18, 64, 29}{54, 39, 66}{68, 71, 75}{29, 7, 42}{7, 40, 21}{10, 53, 19}{76, 87, 57}{7, 3, 82}{77, 35, 30}{51, 26, 53}{44, 72, 22}{58, 87, 80}{75, 16, 46}{61, 64, 63}{70, 80, 26}{81, 26, 37}{22, 56, 19}{50, 47, 52}{18, 41, 86}{75, 40, 9}{29, 31, 48}{52, 82, 19}{45, 39, 12}{33, 36, 27}{73, 54, 20}{14, 71, 74}{72, 3, 68}{28, 55, 10}{23, 78, 73}{31, 41, 30}{59, 54, 12}{7, 72, 50}{26, 73, 48}{15, 78, 20}{42, 24, 52}{34, 70, 0}{83, 33, 4}{82, 37, 46}{15, 33, 43}

{39, 8, 73}{32, 54, 3}{3, 44, 16}{2, 8, 85}{61, 42, 46}{75, 79, 22}{45, 3, 8}{39, 67, 49}{47, 56, 5}{77, 38, 9}{6, 36, 72}{3, 29, 0}{44, 13, 47}{47, 16, 1}{82, 64, 17}{1, 6, 28}{62, 72, 52}{83, 7, 62}{40, 68, 43}{42, 78, 16}{4, 87, 75}{36, 79, 0}{72, 60, 74}{13, 34, 38}{69, 15, 36}{50, 73, 80}{74, 0, 44}{73, 63, 36}{62, 32, 65}{34, 49, 56}{86, 32, 67}{6, 23, 85}{58, 86, 72}{49, 38, 83}{48, 82, 43}{42, 69, 11}{1, 27, 30}{21, 26, 19}{50, 19, 79}{12, 37, 56}{84, 62, 11}{51, 87, 48}{54, 34, 76}{17, 13, 51}{59, 42, 79}

{70, 51, 81}{1, 62, 71}{9, 6, 5}{42, 80, 67}{33, 85, 66}{40, 33, 59}{58, 40, 30}{19, 4, 17}{42, 15, 0}{40, 15, 22}{69, 40, 28}{78, 82, 85}{48, 38, 81}{68, 87, 5}{26, 87, 69}{48, 42, 22}{8, 50, 14}{82, 49, 51}{76, 85, 48}{26, 35, 55}{43, 34, 81}{18, 69, 59}{31, 64, 54}{82, 53, 22}{49, 43, 58}{65, 67, 76}{55, 84, 51}{7, 17, 69}{49, 2, 22}{28, 16, 65}{14, 1, 55}{79, 72, 66}{49, 24, 55}{71, 58, 4}{62, 41, 8}{75, 37, 65}{74, 24, 79}{41, 64, 15}{12, 55, 18}{68, 48, 41}{69, 52, 34}{17, 28, 34}{60, 7, 49}{44, 61, 41}{57, 18, 7}

{0, 67, 13}{87, 65, 78}{70, 76, 59}{54, 63, 83}{53, 71, 6}{61, 16, 39}{41, 22, 83}{54, 74, 56}{18, 63, 4}{73, 6, 47}{43, 56, 60}{79, 48, 28}{73, 44, 56}{78, 52, 57}{61, 79, 57}{21, 48, 25}{85, 15, 25}{34, 8, 46}{52, 3, 41}{57, 86, 63}{72, 77, 14}{87, 0, 33}{75, 63, 49}{25, 75, 60}{29, 15, 28}{73, 64, 5}{55, 60, 50}{28, 18, 72}{36, 70, 65}{7, 76, 33}{35, 25, 58}{43, 72, 78}{35, 60, 80}{70, 16, 23}{34, 87, 22}{50, 43, 24}{47, 41, 75}{57, 15, 35}{29, 50, 63}{16, 67, 73}{46, 31, 66}{10, 81, 0}{20, 10, 1}{30, 83, 55}{20, 35, 86}


{49, 47, 53}{29, 10, 83}{29, 14, 43}{62, 10, 67}{79, 8, 67}{2, 37, 25}{55, 66, 22}{87, 16, 52}{51, 32, 7}{0, 78, 39}{66, 11, 21}{9, 62, 34}{58, 65, 48}{82, 83, 25}{38, 64, 51}{52, 40, 55}{56, 14, 69}{24, 58, 51}{46, 4, 2}{67, 44, 23}{38, 0, 31}{65, 82, 20}{57, 34, 72}{74, 30, 9}{82, 39, 69}{60, 29, 58}{64, 59, 13}{87, 8, 36}{18, 31, 5}{24, 13, 28}{74, 33, 69}{36, 7, 75}{84, 49, 15}{3, 6, 13}{32, 68, 33}{46, 1, 7}{28, 2, 30}{79, 77, 58}{47, 37, 84}{30, 23, 25}{37, 76, 63}{84, 50, 53}{2, 62, 61}{57, 27, 48}{36, 86, 45}

{14, 19, 80}{54, 85, 51}{29, 62, 76}{85, 56, 11}{73, 72, 35}{66, 73, 32}{8, 70, 31}{49, 77, 44}{44, 32, 26}{87, 25, 11}{48, 86, 50}{64, 75, 33}{0, 69, 12}{41, 55, 0}{63, 33, 34}{58, 46, 39}{66, 80, 47}{15, 2, 81}{84, 86, 27}{78, 45, 83}{73, 69, 38}{0, 60, 63}{35, 54, 4}{29, 80, 51}{46, 43, 10}{20, 25, 66}{65, 43, 64}{49, 86, 29}{31, 24, 25}{29, 82, 57}{66, 77, 28}{26, 72, 9}{20, 39, 21}{83, 72, 13}{43, 37, 38}{16, 18, 77}{74, 49, 8}{9, 35, 84}{67, 34, 12}{74, 53, 70}{65, 31, 77}{69, 68, 62}{32, 18, 13}{62, 82, 63}{35, 65, 6}

{82, 36, 30}{38, 41, 72}{9, 76, 3}{16, 49, 76}{9, 86, 47}{27, 53, 52}{84, 72, 5}{26, 16, 41}{70, 41, 79}{43, 76, 45}{79, 44, 37}{53, 63, 14}{25, 70, 69}{42, 38, 65}{13, 57, 10}{5, 10, 71}{86, 1, 61}{22, 24, 5}{31, 60, 10}{45, 15, 82}{59, 61, 87}{1, 40, 29}{48, 67, 37}{53, 81, 83}{77, 54, 87}{38, 45, 26}{59, 5, 49}{40, 13, 62}{44, 6, 40}{17, 54, 18}{12, 49, 64}{64, 84, 57}{43, 85, 47}{50, 44, 87}{52, 56, 30}{23, 32, 49}{58, 6, 69}{47, 46, 35}{0, 46, 84}{84, 58, 81}{37, 57, 40}{24, 45, 33}{34, 32, 53}{41, 36, 53}{55, 62, 45}

{54, 25, 42}{83, 32, 74}{18, 27, 71}{78, 67, 64}{81, 11, 64}{26, 84, 83}{80, 5, 33}{66, 7, 84}{85, 80, 7}{66, 0, 52}{21, 74, 28}{20, 5, 14}{0, 25, 18}{38, 44, 82}{18, 48, 75}{83, 18, 20}{34, 40, 84}{17, 35, 68}{62, 60, 87}{20, 58, 85}{33, 12, 42}{51, 18, 47}{42, 45, 35}{59, 65, 50}{44, 34, 51}{27, 16, 68}{24, 81, 39}{25, 78, 32}{50, 21, 81}{32, 9, 50}{78, 47, 26}{63, 21, 35}{59, 21, 46}{78, 60, 1}{43, 44, 30}{73, 34, 60}{80, 13, 82}{71, 35, 28}{67, 50, 69}{69, 44, 48}{16, 54, 55}{38, 40, 71}{53, 39, 18}{24, 85, 18}{11, 14, 52}


{74, 11, 15}{41, 85, 19}{85, 81, 87}{65, 7, 19}{24, 36, 21}{64, 71, 19}{23, 50, 4}{59, 10, 47}{61, 78, 11}{57, 14, 75}{57, 19, 54}{41, 29, 34}{6, 59, 68}{1, 24, 63}{78, 41, 69}{62, 51, 21}{27, 23, 26}{22, 57, 68}{14, 2, 47}{55, 27, 44}{13, 46, 41}{2, 69, 35}{77, 8, 43}{79, 84, 25}{16, 59, 4}{33, 77, 51}{8, 9, 12}{81, 79, 52}{20, 77, 17}{28, 81, 8}{35, 66, 40}{55, 72, 69}{16, 74, 12}{70, 44, 75}{8, 75, 86}{23, 76, 75}{6, 80, 20}{27, 24, 70}{0, 6, 75}{32, 5, 35}{21, 86, 79}{86, 37, 83}{59, 77, 2}{82, 67, 55}{55, 8, 65}

{63, 69, 32}{61, 15, 26}{10, 79, 76}{8, 51, 52}{60, 14, 9}{11, 37, 9}{20, 81, 72}{45, 28, 31}{30, 50, 15}{60, 51, 30}{6, 2, 29}{81, 23, 80}{26, 11, 29}{34, 47, 27}{40, 63, 51}{12, 19, 24}{3, 40, 14}{27, 41, 5}{39, 13, 9}{68, 55, 58}{26, 4, 1}{81, 16, 22}{2, 5, 44}{14, 18, 37}{37, 55, 17}{85, 86, 68}{12, 87, 82}{67, 46, 26}{18, 36, 9}{53, 66, 86}{38, 85, 75}{66, 48, 1}{35, 52, 7}{57, 8, 38}{37, 8, 35}{50, 83, 1}{39, 33, 11}{54, 26, 52}{64, 22, 60}{14, 64, 36}{21, 10, 16}{53, 40, 67}{36, 26, 22}{20, 61, 8}{64, 3, 69}

{39, 28, 14}{79, 11, 20}{29, 74, 87}{60, 2, 40}{60, 16, 37}{11, 47, 76}{51, 57, 46}{68, 26, 30}{27, 80, 2}{1, 11, 44}{45, 47, 60}{81, 62, 56}{65, 83, 46}{86, 33, 82}{20, 74, 7}{65, 56, 68}{5, 79, 54}{71, 70, 37}{57, 71, 59}{74, 6, 19}{47, 29, 8}{11, 17, 6}{19, 34, 45}{23, 1, 77}{45, 64, 27}{31, 52, 37}{56, 29, 4}{65, 54, 80}{52, 74, 5}{2, 87, 32}{50, 56, 76}{29, 30, 75}{54, 41, 28}{10, 78, 36}{12, 17, 14}{77, 32, 70}{1, 75, 52}{25, 55, 19}{71, 69, 16}{27, 81, 77}{0, 71, 21}{17, 3, 50}{26, 86, 25}{20, 70, 3}{22, 20, 23}

{14, 10, 65}{61, 81, 40}{23, 52, 9}{46, 81, 12}{17, 74, 48}{75, 62, 66}{86, 17, 87}{28, 22, 61}{71, 20, 49}{47, 4, 24}{15, 60, 24}{47, 21, 65}{69, 19, 60}{14, 84, 31}{35, 41, 82}{35, 29, 39}{50, 78, 12}{76, 5, 86}{83, 23, 66}{54, 7, 11}{45, 50, 22}{48, 61, 4}{10, 68, 51}{63, 77, 84}{37, 62, 59}{75, 39, 42}{56, 17, 79}{14, 32, 45}{60, 70, 57}{10, 4, 72}{42, 56, 51}{67, 57, 4}{70, 4, 82}{24, 69, 75}{31, 75, 21}{84, 56, 78}{18, 30, 8}{46, 23, 5}{4, 43, 41}{7, 44, 25}{67, 38, 36}{76, 32, 30}{70, 52, 58}{79, 60, 6}{67, 60, 54}


{83, 85, 16}{31, 82, 16}{57, 50, 16}{18, 81, 60}{2, 57, 24}{61, 47, 54}{53, 24, 17}{41, 10, 77}{35, 14, 44}{6, 82, 21}{50, 41, 20}{83, 5, 48}{43, 42, 70}{46, 44, 9}{56, 21, 57}{28, 27, 42}{30, 37, 64}{45, 70, 84}{15, 65, 51}{33, 48, 10}{30, 21, 42}{50, 46, 33}{63, 68, 25}{1, 13, 42}{80, 4, 39}{86, 74, 65}{0, 83, 47}{22, 3, 18}{32, 57, 47}{58, 9, 15}{52, 45, 48}{74, 38, 63}{5, 39, 51}{66, 71, 13}{20, 27, 46}{58, 23, 13}{27, 22, 25}{62, 28, 43}{32, 17, 21}{85, 4, 14}{56, 55, 86}{86, 24, 11}{43, 26, 12}{44, 53, 57}{85, 67, 28}

{55, 4, 53}{15, 32, 38}{19, 47, 81}{65, 53, 30}{11, 22, 80}{6, 52, 15}{13, 2, 76}{77, 82, 56}{18, 56, 45}{35, 64, 74}{79, 30, 16}{17, 84, 85}{20, 2, 16}{73, 0, 43}{85, 34, 30}{43, 61, 84}{3, 65, 63}{57, 42, 83}{23, 10, 69}{63, 22, 52}{28, 19, 5}{76, 58, 53}{77, 34, 6}{15, 8, 19}{65, 66, 24}{40, 77, 47}{16, 84, 6}{51, 9, 69}{80, 53, 15}{56, 13, 36}{3, 74, 36}{47, 62, 74}{10, 25, 8}{71, 36, 50}{59, 1, 39}{24, 10, 35}{28, 50, 51}{38, 2, 79}{10, 56, 3}{83, 36, 40}{4, 37, 32}{26, 28, 64}{28, 56, 33}{78, 80, 37}{11, 38, 12}

{26, 59, 85}{61, 49, 35}{73, 55, 76}{38, 55, 5}{63, 81, 67}{14, 48, 13}{61, 70, 19}{53, 11, 2}{33, 52, 21}{39, 10, 27}{9, 68, 45}{70, 33, 2}{16, 33, 29}{34, 20, 31}{12, 29, 66}{58, 37, 27}{60, 27, 21}{45, 65, 72}{74, 13, 75}{23, 45, 51}{23, 38, 29}{48, 30, 12}{81, 55, 36}{71, 24, 67}{29, 71, 65}{58, 38, 17}{77, 11, 0}{6, 4, 8}{14, 73, 82}{46, 60, 77}{28, 87, 83}{63, 9, 28}{0, 49, 26}{9, 78, 24}{10, 54, 84}{83, 34, 14}{14, 66, 81}{37, 36, 23}{20, 13, 19}{36, 47, 48}{39, 57, 26}{83, 15, 68}{9, 16, 19}{9, 67, 21}{37, 66, 49}

{79, 53, 64}{63, 48, 11}{45, 44, 80}{19, 36, 39}{8, 78, 5}{24, 6, 7}{58, 75, 78}{86, 4, 3}{0, 59, 20}{33, 53, 54}{20, 9, 53}{58, 21, 14}{38, 28, 59}{12, 5, 63}{48, 62, 20}{43, 7, 9}{84, 42, 3}{3, 55, 2}{48, 3, 53}{68, 61, 50}{72, 19, 1}{43, 5, 25}{69, 84, 65}{29, 36, 54}{30, 39, 72}{49, 45, 40}{53, 87, 43}{45, 79, 4}{60, 32, 82}{33, 37, 72}{12, 3, 23}{86, 44, 71}{22, 13, 84}{80, 9, 83}{68, 31, 78}{13, 86, 12}{65, 39, 52}{2, 0, 86}{15, 72, 27}{30, 4, 73}{42, 17, 47}{76, 72, 31}{62, 26, 79}{61, 14, 23}{50, 49, 13}


{13, 11, 55}{66, 59, 56}{29, 70, 55}{33, 55, 78}{71, 84, 48}{71, 22, 51}{76, 35, 0}{84, 80, 30}{0, 65, 61}{54, 49, 27}{66, 30, 45}{81, 69, 86}{32, 36, 42}{10, 12, 40}{65, 4, 22}{49, 78, 28}{86, 64, 34}{22, 39, 37}{87, 27, 9}{44, 64, 58}{1, 43, 2}{23, 72, 59}{25, 67, 45}{84, 73, 75}{14, 49, 68}{32, 41, 12}{25, 59, 14}{12, 75, 53}{49, 18, 11}{76, 18, 80}{7, 86, 59}{68, 0, 53}{18, 19, 73}{25, 13, 52}{13, 35, 31}{63, 56, 58}{38, 47, 68}{20, 45, 87}{46, 40, 74}{57, 85, 62}{72, 61, 51}{86, 19, 42}{64, 46, 52}{7, 77, 48}{47, 25, 3}

{87, 30, 3}{36, 1, 80}{6, 31, 57}{79, 83, 69}{50, 6, 64}{81, 68, 29}{38, 76, 19}{41, 37, 74}{7, 13, 16}{73, 46, 28}{3, 61, 33}{78, 18, 35}{57, 11, 30}{85, 9, 55}{34, 79, 65}{58, 16, 62}{11, 82, 72}{68, 37, 24}{59, 44, 81}{31, 73, 42}{31, 43, 22}{82, 75, 61}{86, 23, 28}{86, 16, 51}{28, 37, 0}{81, 74, 76}{79, 29, 9}{14, 33, 79}{82, 79, 27}{51, 0, 1}{56, 25, 6}{0, 5, 58}{28, 75, 80}{2, 67, 52}{81, 30, 7}{25, 61, 12}{1, 82, 5}{67, 17, 61}{30, 71, 33}{62, 15, 18}{46, 17, 29}{44, 85, 39}{10, 6, 37}{62, 73, 24}{38, 25, 16}

{49, 10, 30}{22, 73, 85}{15, 4, 13}{4, 69, 31}{19, 49, 48}{11, 45, 46}{21, 1, 12}{31, 49, 62}{56, 71, 41}{5, 70, 50}{43, 6, 39}{48, 35, 23}{54, 37, 50}{41, 66, 67}{14, 15, 16}{63, 46, 19}{55, 21, 80}{15, 10, 86}{8, 76, 1}{32, 1, 22}{25, 4, 51}{48, 55, 46}{16, 34, 5}{84, 18, 87}{75, 55, 34}{0, 57, 45}{77, 64, 4}{15, 56, 46}{26, 5, 40}{35, 33, 22}{38, 24, 84}{87, 41, 40}{10, 64, 85}{66, 61, 6}{35, 53, 16}{43, 32, 79}{61, 52, 18}{22, 10, 9}{79, 13, 68}{27, 56, 7}{9, 71, 42}{80, 77, 3}{27, 4, 74}{39, 86, 60}{71, 78, 3}

{54, 23, 24}{31, 32, 81}{76, 24, 61}{62, 80, 17}{55, 43, 74}{35, 85, 70}{8, 84, 82}{28, 58, 32}{56, 1, 70}{63, 41, 6}{39, 2, 48}{1, 35, 34}{6, 49, 87}{64, 87, 1}{32, 71, 85}{79, 46, 80}{80, 57, 43}{87, 13, 70}{4, 34, 21}{15, 48, 70}{75, 15, 5}{13, 27, 73}{76, 6, 51}{74, 45, 1}{7, 34, 37}{76, 82, 40}{46, 76, 69}{56, 38, 39}{17, 30, 5}{76, 83, 21}{30, 13, 63}{12, 72, 47}{59, 84, 29}{39, 76, 41}{59, 34, 15}{81, 75, 54}{23, 40, 19}{74, 51, 78}{5, 66, 60}{42, 77, 68}{24, 44, 83}{2, 84, 23}{11, 36, 58}{62, 23, 0}{52, 38, 80}


{58, 8, 33}{55, 32, 61}{57, 58, 12}{49, 80, 69}{65, 40, 85}{42, 62, 64}{12, 7, 22}{46, 3, 85}{55, 64, 20}

{7, 73, 70}{52, 43, 86}{26, 71, 76}{1, 53, 31}{31, 40, 27}{21, 73, 2}{62, 19, 44}{11, 70, 28}{85, 12, 27}

{52, 59, 32}{2, 68, 19}{46, 72, 53}{78, 66, 17}{71, 61, 60}{56, 61, 9}{75, 2, 56}{70, 66, 9}{78, 63, 2}

{22, 47, 58}{38, 21, 3}{24, 59, 30}{36, 31, 59}{23, 11, 8}{39, 70, 68}{42, 49, 72}


g = 5, u = 10:

{8, 32, 15}{37, 9, 18}{27, 38, 20}{20, 19, 36}{9, 6, 21}{31, 3, 17}{20, 34, 2}{29, 33, 20}{24, 6, 38}{35, 1, 43}{39, 22, 30}{40, 38, 12}{34, 28, 19}{15, 6, 0}{37, 14, 5}{27, 14, 26}{16, 13, 45}{2, 31, 25}{3, 2, 27}{39, 26, 32}{18, 17, 40}{40, 39, 14}{22, 26, 49}{48, 6, 23}{7, 14, 2}{41, 48, 29}{21, 37, 13}{30, 29, 44}{31, 30, 35}{17, 2, 45}{17, 8, 25}{3, 49, 15}{45, 29, 6}{14, 33, 16}{28, 46, 45}{39, 4, 20}{22, 40, 21}{43, 14, 8}{48, 3, 22}{41, 42, 4}{5, 9, 26}{13, 18, 27}{7, 16, 48}{49, 24, 31}{28, 23, 9}

{37, 11, 15}{19, 15, 30}{22, 43, 34}{29, 16, 5}{27, 29, 34}{38, 33, 35}{11, 34, 16}{20, 9, 42}{25, 47, 12}{21, 35, 44}{5, 0, 32}{3, 41, 16}{24, 35, 16}{36, 12, 0}{30, 47, 38}{20, 23, 15}{0, 46, 33}{10, 41, 24}{24, 48, 39}{35, 22, 19}{13, 25, 9}{29, 11, 24}{40, 34, 9}{25, 24, 1}{36, 1, 13}{3, 6, 44}{7, 15, 41}{43, 28, 41}{4, 12, 23}{21, 30, 25}{4, 21, 43}{18, 30, 46}{27, 16, 23}{16, 1, 22}{16, 8, 39}{10, 25, 39}{26, 29, 23}{17, 28, 42}{20, 41, 46}{43, 18, 26}{4, 48, 15}{2, 10, 11}{6, 31, 7}{7, 36, 29}{5, 34, 31}

{12, 34, 10}{3, 1, 4}{9, 4, 16}{3, 34, 7}{19, 1, 37}{28, 15, 27}{31, 47, 14}{7, 42, 49}{8, 4, 0}{22, 47, 20}{16, 21, 28}{39, 5, 21}{17, 33, 22}{44, 45, 49}{30, 1, 32}{21, 46, 49}{7, 44, 43}{13, 6, 32}{11, 5, 48}{10, 18, 36}{4, 29, 17}{17, 6, 1}{44, 19, 2}{3, 36, 24}{12, 14, 15}{20, 21, 48}{40, 29, 2}{19, 32, 18}{8, 34, 30}{13, 17, 26}{30, 3, 42}{32, 23, 21}{49, 0, 38}{31, 36, 9}{19, 3, 5}{23, 47, 35}{9, 8, 47}{12, 5, 6}{43, 25, 49}{0, 31, 37}{23, 17, 5}{3, 18, 0}{29, 0, 25}{28, 39, 37}{22, 8, 27}

{17, 35, 9}{49, 13, 41}{13, 5, 8}{0, 19, 26}{39, 47, 6}{25, 33, 19}{0, 7, 39}{49, 47, 1}{15, 1, 9}{32, 29, 35}{22, 7, 38}{37, 23, 42}{47, 0, 21}{7, 46, 23}{31, 43, 10}{11, 0, 13}{7, 28, 25}{16, 19, 38}{30, 9, 2}{18, 45, 14}{35, 40, 4}{40, 37, 43}{15, 36, 40}{3, 45, 39}{40, 47, 48}{7, 40, 5}{17, 43, 32}{7, 32, 11}{2, 24, 13}{11, 22, 46}{47, 3, 28}{15, 17, 38}{22, 9, 41}{1, 5, 27}{20, 43, 5}{14, 20, 28}{29, 42, 1}{27, 42, 35}{10, 33, 9}{44, 46, 38}{22, 25, 4}{28, 13, 44}{8, 26, 42}{5, 4, 47}{21, 29, 18}


{13, 46, 40}{33, 44, 5}{25, 42, 18}{20, 44, 1}{11, 8, 44}{6, 34, 18}{6, 42, 11}{4, 38, 13}{16, 49, 18}{43, 39, 38}{45, 33, 1}{40, 28, 1}{35, 11, 20}{1, 39, 2}{22, 23, 0}{27, 9, 43}{4, 26, 33}{7, 10, 4}{48, 2, 35}{27, 24, 33}{38, 32, 45}{13, 12, 31}{44, 18, 22}{37, 16, 20}{34, 42, 13}{25, 44, 16}{17, 34, 0}{6, 10, 35}{32, 4, 27}{36, 2, 4}{40, 41, 23}{32, 25, 41}{12, 26, 20}{22, 36, 5}{24, 45, 47}{29, 28, 12}{46, 24, 5}{2, 33, 28}{10, 13, 29}{42, 14, 36}{5, 30, 41}{15, 13, 39}{19, 40, 31}{13, 35, 14}{3, 29, 14}

{35, 41, 8}{27, 45, 31}{32, 49, 34}{11, 25, 3}{19, 6, 43}{48, 19, 27}{12, 30, 45}{49, 4, 30}{2, 15, 21}{10, 5, 28}{26, 2, 41}{37, 25, 48}{17, 10, 19}{8, 23, 31}{22, 37, 29}{11, 36, 30}{39, 18, 35}{14, 46, 9}{1, 7, 26}{25, 36, 23}{32, 36, 28}{11, 17, 49}{0, 28, 24}{36, 33, 49}{28, 49, 35}{47, 15, 18}{35, 34, 46}{48, 17, 46}{8, 29, 46}{7, 33, 30}{46, 25, 27}{4, 19, 46}{40, 45, 26}{14, 23, 30}{46, 39, 42}{38, 3, 10}{0, 1, 14}{17, 44, 36}{1, 10, 46}{3, 40, 8}{15, 29, 43}{14, 49, 48}{24, 21, 42}{12, 7, 35}{36, 39, 27}

{12, 1, 8}{34, 36, 47}{31, 38, 29}{15, 24, 22}{3, 35, 26}{44, 23, 10}{9, 0, 45}{47, 43, 16}{12, 9, 3}{30, 17, 16}{14, 10, 32}{43, 45, 11}{30, 24, 43}{16, 10, 15}{23, 34, 45}{15, 33, 42}{21, 7, 45}{24, 23, 19}{37, 10, 26}{39, 31, 33}{45, 20, 8}{48, 12, 33}{28, 6, 4}{23, 2, 38}{25, 26, 34}{21, 19, 14}{8, 36, 21}{11, 40, 33}{1, 23, 18}{19, 11, 12}{3, 32, 20}{42, 10, 47}{44, 31, 42}{11, 39, 23}{38, 25, 14}{0, 43, 42}{37, 32, 44}{8, 24, 37}{33, 6, 37}{6, 8, 2}{2, 46, 47}{30, 27, 6}{48, 44, 0}{40, 49, 6}{12, 18, 24}

{39, 41, 34}{26, 31, 48}{21, 34, 33}{4, 37, 45}{27, 49, 12}{48, 30, 13}{36, 43, 48}{45, 22, 10}{49, 10, 8}{37, 3, 46}{13, 20, 7}{38, 26, 21}{48, 1, 34}{9, 48, 32}{0, 16, 2}{40, 42, 16}{26, 15, 44}{47, 26, 11}{7, 9, 24}{41, 33, 18}{19, 41, 45}{37, 36, 35}{26, 28, 30}{41, 17, 14}{17, 21, 12}{47, 44, 41}{27, 44, 40}{19, 47, 13}{15, 31, 46}{24, 32, 40}{12, 44, 39}{4, 18, 11}{33, 32, 47}{42, 45, 48}{41, 36, 38}{12, 41, 37}{19, 7, 8}{27, 41, 0}{31, 18, 20}{5, 42, 38}{34, 37, 38}{20, 24, 17}{9, 11, 38}{27, 21, 10}{22, 14, 6}


{22, 28, 31}{49, 37, 2}

{16, 31, 32}{12, 43, 46}

{2, 18, 5}{6, 20, 25}


g = 5, u = 14:

{54, 64, 57}{60, 40, 49}{43, 59, 44}{41, 58, 47}{14, 37, 53}{20, 61, 4}{41, 31, 65}{57, 61, 42}{33, 14, 24}{33, 62, 64}{27, 33, 3}{32, 13, 57}{49, 44, 17}{9, 24, 1}{54, 34, 19}{69, 67, 29}{62, 61, 17}{53, 3, 49}{10, 33, 34}{22, 18, 21}{5, 62, 67}{29, 5, 56}{39, 41, 68}{44, 51, 18}{64, 2, 29}{63, 51, 41}{55, 31, 66}{28, 55, 19}{65, 64, 25}{54, 67, 43}{20, 28, 30}{58, 23, 4}{1, 66, 26}{53, 29, 41}{17, 37, 12}{69, 17, 8}{69, 42, 39}{68, 16, 19}{64, 24, 61}{21, 11, 10}{63, 2, 47}{32, 25, 9}{14, 60, 65}{9, 16, 33}{5, 14, 50}

{20, 7, 66}{22, 67, 27}{24, 0, 11}{55, 1, 46}{37, 38, 69}{8, 61, 32}{62, 28, 69}{59, 61, 58}{55, 47, 12}{1, 39, 22}{19, 20, 13}{6, 43, 21}{25, 26, 33}{62, 43, 39}{9, 36, 66}{19, 31, 67}{6, 12, 42}{47, 25, 16}{8, 37, 26}{52, 68, 20}{58, 36, 45}{57, 19, 40}{54, 56, 61}{23, 40, 64}{0, 9, 64}{21, 33, 52}{47, 6, 14}{21, 19, 53}{14, 27, 58}{26, 56, 27}{38, 4, 3}{22, 57, 17}{1, 20, 0}{15, 49, 48}{13, 0, 54}{16, 59, 64}{53, 52, 45}{36, 60, 42}{45, 67, 20}{15, 28, 12}{52, 56, 62}{8, 63, 54}{64, 42, 46}{22, 12, 69}{4, 36, 15}

{35, 37, 2}{56, 34, 64}{5, 8, 1}{12, 4, 57}{60, 10, 9}{35, 47, 44}{0, 63, 65}{0, 50, 23}{32, 11, 1}{23, 31, 54}{12, 20, 43}{29, 30, 47}{59, 32, 56}{48, 40, 0}{10, 4, 55}{3, 24, 62}{47, 20, 53}{28, 54, 52}{35, 0, 45}{38, 28, 46}{63, 17, 19}{67, 1, 17}{36, 38, 34}{46, 16, 54}{8, 15, 16}{35, 20, 46}{12, 60, 53}{56, 36, 63}{35, 32, 23}{38, 5, 6}{32, 42, 30}{53, 24, 57}{28, 67, 10}{66, 19, 8}{5, 31, 16}{22, 56, 19}{61, 69, 43}{24, 35, 36}{5, 44, 23}{51, 43, 7}{21, 65, 34}{22, 32, 54}{42, 25, 20}{43, 41, 49}{68, 25, 10}

{40, 39, 32}{25, 61, 60}{68, 37, 6}{34, 63, 45}{39, 15, 45}{9, 26, 22}{0, 67, 36}{52, 55, 7}{4, 59, 5}{9, 54, 69}{34, 31, 69}{9, 8, 7}{53, 44, 68}{29, 35, 10}{50, 28, 17}{44, 36, 21}{23, 22, 41}{42, 22, 68}{16, 42, 62}{13, 9, 30}{13, 31, 36}{25, 21, 13}{44, 6, 67}{57, 69, 7}{11, 30, 12}{34, 58, 1}{10, 22, 48}{44, 54, 65}{50, 65, 46}{23, 24, 25}{18, 3, 57}{55, 58, 63}{15, 55, 25}{9, 5, 63}{49, 22, 24}{45, 33, 37}{67, 13, 46}{49, 26, 28}{66, 22, 2}{10, 50, 54}{17, 65, 55}{51, 22, 3}{66, 34, 53}{18, 31, 58}{33, 15, 44}


{42, 34, 47}{58, 20, 9}{65, 11, 68}{44, 19, 9}{48, 21, 14}{18, 33, 13}{37, 4, 54}{16, 56, 13}{65, 57, 20}{47, 8, 38}{65, 16, 10}{19, 4, 14}{0, 21, 3}{44, 25, 1}{60, 0, 16}{14, 66, 54}{57, 45, 62}{49, 25, 51}{30, 15, 65}{14, 30, 39}{64, 53, 1}{26, 29, 45}{19, 2, 41}{22, 58, 6}{18, 52, 23}{24, 29, 21}{14, 20, 2}{24, 17, 7}{19, 26, 52}{40, 7, 2}{31, 25, 7}{18, 28, 65}{9, 12, 14}{64, 13, 11}{47, 45, 32}{9, 55, 21}{44, 31, 11}{32, 34, 15}{13, 62, 29}{67, 38, 68}{39, 17, 52}{32, 63, 38}{5, 13, 58}{2, 33, 48}{68, 62, 59}

{67, 16, 7}{40, 17, 5}{63, 18, 25}{6, 29, 19}{10, 44, 64}{3, 55, 11}{33, 60, 22}{18, 26, 11}{55, 18, 30}{30, 19, 64}{17, 60, 29}{6, 9, 46}{31, 14, 29}{10, 15, 14}{67, 9, 52}{65, 26, 59}{39, 19, 7}{50, 39, 29}{42, 53, 50}{28, 24, 44}{47, 50, 68}{51, 34, 8}{49, 47, 10}{11, 40, 56}{35, 22, 59}{48, 65, 67}{48, 51, 27}{10, 31, 26}{29, 68, 8}{39, 57, 44}{64, 17, 43}{27, 30, 21}{30, 69, 59}{4, 27, 6}{38, 25, 40}{15, 5, 60}{54, 29, 20}{44, 55, 40}{61, 66, 11}{43, 55, 23}{15, 47, 0}{5, 30, 10}{21, 5, 42}{0, 4, 44}{5, 48, 3}

{21, 17, 16}{39, 27, 47}{20, 51, 60}{67, 30, 63}{7, 6, 0}{13, 50, 2}{55, 2, 53}{12, 45, 8}{23, 34, 12}{60, 34, 30}{56, 48, 43}{48, 8, 53}{42, 45, 10}{52, 2, 65}{35, 11, 19}{0, 19, 59}{46, 33, 68}{40, 50, 21}{66, 65, 40}{12, 29, 7}{33, 36, 30}{23, 20, 17}{38, 50, 9}{39, 6, 66}{25, 35, 12}{40, 9, 47}{1, 42, 33}{5, 65, 53}{43, 5, 34}{38, 13, 39}{66, 47, 48}{49, 13, 52}{3, 40, 10}{61, 36, 12}{40, 51, 16}{23, 27, 68}{6, 33, 41}{66, 25, 37}{8, 24, 46}{49, 54, 45}{1, 63, 23}{64, 60, 31}{17, 0, 34}{37, 24, 56}{8, 13, 65}

{65, 27, 62}{68, 34, 18}{0, 2, 39}{53, 22, 63}{55, 64, 6}{65, 33, 43}{32, 68, 5}{12, 19, 50}{51, 5, 66}{4, 29, 9}{4, 13, 26}{54, 58, 17}{40, 4, 41}{66, 16, 4}{59, 54, 55}{51, 38, 57}{16, 37, 61}{51, 10, 69}{17, 26, 35}{58, 24, 50}{18, 15, 59}{23, 3, 36}{52, 48, 61}{22, 0, 46}{62, 58, 60}{58, 38, 29}{23, 59, 57}{31, 62, 8}{51, 1, 56}{44, 46, 34}{7, 61, 26}{14, 63, 61}{3, 29, 52}{24, 19, 51}{9, 42, 27}{60, 66, 50}{30, 68, 57}{10, 8, 6}{7, 56, 38}{39, 31, 56}{9, 35, 39}{46, 47, 17}{22, 55, 61}{52, 32, 6}{11, 8, 27}


{66, 44, 56}{11, 69, 33}{0, 41, 5}{23, 47, 7}{44, 22, 37}{61, 29, 44}{63, 26, 16}{45, 18, 5}{67, 47, 4}{31, 42, 63}{0, 25, 62}{62, 19, 36}{4, 63, 64}{65, 7, 22}{28, 13, 22}{56, 18, 47}{69, 46, 36}{60, 37, 55}{18, 40, 1}{67, 3, 60}{36, 5, 55}{17, 4, 68}{43, 36, 28}{60, 45, 19}{23, 16, 49}{39, 10, 58}{13, 37, 15}{33, 56, 55}{19, 58, 46}{14, 41, 3}{9, 48, 59}{23, 21, 38}{53, 7, 46}{56, 69, 23}{61, 27, 46}{22, 5, 52}{35, 60, 38}{63, 60, 43}{46, 39, 5}{20, 22, 15}{6, 25, 56}{61, 10, 53}{26, 60, 6}{50, 63, 59}{62, 14, 40}

{9, 2, 43}{49, 31, 27}{43, 40, 22}{36, 48, 57}{53, 6, 51}{36, 59, 49}{31, 40, 20}{32, 66, 69}{38, 42, 44}{11, 47, 37}{0, 53, 30}{39, 54, 36}{64, 39, 48}{22, 14, 11}{11, 29, 36}{36, 20, 27}{18, 36, 10}{11, 23, 42}{45, 4, 1}{50, 37, 30}{41, 60, 52}{46, 48, 30}{68, 28, 3}{43, 3, 35}{17, 42, 51}{68, 7, 60}{58, 35, 65}{49, 42, 37}{30, 8, 52}{35, 14, 1}{42, 66, 35}{16, 38, 41}{18, 54, 48}{65, 3, 42}{57, 60, 28}{44, 60, 27}{39, 59, 60}{66, 67, 57}{34, 41, 24}{0, 29, 51}{16, 20, 55}{4, 11, 49}{31, 38, 2}{15, 69, 35}{28, 11, 41}

{35, 18, 61}{62, 23, 10}{2, 3, 8}{60, 47, 24}{32, 51, 55}{14, 51, 52}{13, 43, 42}{35, 40, 13}{6, 54, 3}{5, 37, 57}{49, 68, 9}{51, 54, 11}{43, 26, 30}{24, 13, 6}{24, 32, 48}{18, 69, 6}{24, 43, 31}{3, 30, 66}{10, 1, 37}{21, 39, 61}{4, 33, 35}{0, 68, 43}{31, 46, 51}{1, 60, 13}{2, 61, 28}{25, 14, 43}{7, 41, 18}{5, 7, 11}{29, 23, 48}{8, 35, 41}{35, 31, 57}{33, 32, 28}{26, 41, 20}{18, 0, 66}{39, 34, 37}{37, 67, 40}{0, 55, 8}{44, 7, 13}{0, 10, 27}{64, 28, 51}{67, 34, 59}{56, 12, 58}{55, 24, 68}{6, 31, 61}{18, 17, 2}

{69, 20, 3}{15, 38, 11}{13, 47, 3}{50, 16, 43}{32, 41, 64}{38, 64, 27}{50, 61, 41}{14, 16, 18}{64, 67, 58}{67, 12, 33}{15, 50, 52}{2, 26, 34}{18, 20, 64}{51, 45, 68}{5, 12, 49}{35, 52, 34}{47, 64, 69}{15, 9, 41}{23, 28, 53}{3, 61, 1}{11, 62, 46}{62, 4, 7}{12, 46, 2}{69, 49, 0}{53, 32, 43}{47, 26, 51}{48, 42, 4}{8, 58, 57}{38, 33, 17}{52, 11, 57}{26, 0, 57}{69, 26, 5}{28, 48, 63}{16, 45, 24}{34, 11, 16}{38, 14, 26}{43, 45, 38}{40, 28, 29}{33, 66, 23}{12, 41, 66}{30, 38, 54}{32, 16, 3}{26, 64, 3}{28, 47, 31}{39, 65, 24}


{65, 19, 69}{16, 29, 22}{24, 59, 40}{27, 17, 15}{27, 12, 24}{45, 48, 55}{46, 45, 66}{53, 56, 17}{32, 44, 50}{50, 11, 20}{1, 31, 50}{56, 65, 4}{34, 50, 3}{12, 38, 65}{61, 67, 49}{12, 62, 63}{25, 52, 36}{29, 63, 66}{34, 22, 4}{47, 21, 57}{20, 37, 63}{68, 63, 13}{62, 22, 38}{61, 40, 15}{60, 56, 8}{59, 20, 38}{64, 45, 14}{68, 1, 36}{54, 53, 15}{44, 12, 3}{26, 53, 58}{12, 1, 16}{57, 25, 27}{58, 11, 43}{68, 58, 15}{65, 29, 32}{36, 65, 47}{21, 67, 8}{35, 16, 48}{45, 22, 25}{49, 20, 39}{2, 32, 67}{16, 27, 53}{46, 25, 41}{3, 46, 15}

{61, 34, 9}{49, 18, 8}{30, 31, 22}{36, 6, 17}{59, 41, 10}{26, 36, 32}{11, 60, 48}{35, 62, 53}{37, 28, 27}{38, 48, 1}{52, 16, 69}{12, 39, 18}{10, 17, 13}{28, 4, 25}{52, 64, 37}{69, 40, 63}{59, 66, 13}{33, 49, 29}{2, 5, 27}{32, 27, 19}{3, 19, 37}{52, 12, 0}{36, 51, 2}{54, 47, 62}{56, 9, 3}{43, 8, 4}{14, 68, 69}{44, 8, 14}{59, 1, 47}{41, 62, 44}{66, 27, 43}{21, 69, 1}{64, 66, 49}{2, 23, 60}{23, 19, 15}{50, 6, 45}{19, 38, 49}{21, 20, 32}{8, 40, 42}{51, 58, 21}{20, 8, 33}{47, 43, 52}{13, 23, 45}{68, 66, 21}{32, 12, 31}

{35, 55, 67}{28, 1, 6}{67, 14, 23}{46, 63, 10}{45, 28, 9}{29, 46, 37}{62, 9, 18}{15, 67, 51}{52, 44, 63}{33, 51, 50}{37, 43, 18}{21, 26, 62}{49, 2, 6}{21, 12, 64}{1, 54, 41}{23, 26, 46}{28, 34, 7}{59, 27, 29}{48, 68, 31}{21, 31, 37}{62, 49, 30}{5, 20, 24}{3, 63, 39}{56, 45, 30}{41, 42, 67}{4, 51, 39}{41, 57, 56}{53, 33, 59}{40, 36, 53}{58, 37, 0}{33, 40, 58}{30, 1, 7}{57, 46, 14}{11, 45, 2}{14, 13, 34}{59, 2, 25}{62, 2, 1}{39, 55, 26}{59, 46, 21}{68, 56, 2}{6, 30, 40}{39, 16, 28}{45, 41, 21}{57, 49, 50}{54, 42, 24}

{32, 37, 62}{68, 35, 64}{18, 67, 24}{7, 32, 10}{46, 40, 52}{2, 42, 15}{29, 34, 25}{49, 1, 65}{52, 42, 59}{49, 56, 46}{16, 57, 6}{26, 15, 24}{7, 33, 54}{18, 27, 50}{45, 65, 61}{26, 48, 44}{39, 8, 23}{14, 17, 32}{28, 8, 59}{51, 12, 59}{9, 11, 17}{57, 55, 34}{60, 21, 54}{32, 58, 49}{61, 23, 30}{42, 18, 19}{10, 20, 56}{50, 26, 67}{57, 33, 63}{25, 50, 69}{52, 27, 1}{28, 5, 35}{25, 30, 17}{7, 14, 36}{58, 3, 25}{35, 27, 54}{13, 48, 12}{38, 18, 53}{33, 31, 0}{28, 66, 58}{30, 51, 35}{56, 15, 21}{15, 64, 7}{69, 24, 2}{44, 69, 45}


{37, 36, 41}{63, 15, 6}{19, 10, 43}{48, 41, 17}{59, 11, 6}{4, 31, 52}{55, 50, 62}

{53, 31, 9}{50, 35, 56}{57, 10, 2}{5, 25, 54}{55, 42, 29}{59, 7, 37}{61, 13, 51}

{34, 40, 27}{45, 7, 27}{7, 42, 58}{38, 61, 0}{4, 53, 69}{25, 19, 48}{21, 4, 2}

{14, 55, 49}{30, 4, 24}{58, 69, 48}{48, 7, 50}{15, 66, 62}


g = 5, u = 16:

{44, 34, 55}{44, 10, 29}{20, 75, 65}{41, 67, 22}{29, 63, 73}{1, 43, 23}{53, 22, 36}{69, 40, 23}{20, 63, 51}{61, 63, 62}{46, 37, 23}{16, 33, 69}{50, 0, 58}{7, 77, 59}{11, 46, 10}{41, 79, 11}{17, 50, 76}{69, 10, 7}{33, 60, 15}{11, 58, 33}{31, 6, 35}{47, 26, 17}{10, 16, 63}{11, 2, 23}{79, 6, 12}{79, 22, 50}{31, 37, 58}{5, 34, 29}{24, 18, 45}{4, 35, 37}{12, 22, 34}{43, 58, 62}{71, 62, 25}{59, 72, 17}{79, 46, 74}{59, 10, 57}{36, 12, 75}{4, 31, 51}{64, 73, 44}{56, 39, 31}{76, 55, 30}{48, 19, 77}{38, 78, 52}{13, 64, 67}{43, 48, 72}

{20, 5, 32}{7, 62, 53}{12, 14, 16}{39, 41, 58}{8, 76, 51}{51, 30, 64}{47, 23, 3}{22, 47, 2}{31, 12, 52}{42, 64, 15}{64, 17, 36}{31, 33, 68}{29, 42, 22}{21, 18, 8}{39, 34, 9}{17, 28, 70}{67, 1, 26}{16, 2, 7}{20, 74, 54}{26, 65, 69}{13, 11, 52}{68, 22, 17}{45, 48, 41}{5, 79, 56}{35, 44, 14}{23, 57, 77}{28, 79, 25}{36, 66, 67}{72, 4, 50}{68, 72, 10}{34, 48, 46}{14, 33, 38}{56, 17, 20}{73, 39, 76}{58, 67, 29}{14, 17, 8}{39, 65, 42}{70, 7, 24}{75, 61, 22}{40, 76, 11}{30, 35, 11}{3, 57, 56}{51, 32, 53}{39, 26, 51}{36, 18, 37}

{56, 45, 11}{73, 18, 6}{3, 44, 30}{22, 58, 4}{41, 26, 77}{73, 50, 35}{41, 40, 32}{10, 8, 23}{2, 58, 5}{2, 12, 25}{17, 60, 37}{17, 3, 43}{77, 16, 56}{29, 20, 3}{57, 69, 67}{25, 10, 5}{45, 42, 78}{64, 10, 55}{39, 57, 64}{55, 32, 2}{73, 33, 10}{63, 72, 46}{78, 16, 21}{4, 10, 77}{53, 23, 42}{37, 75, 51}{17, 40, 38}{18, 68, 76}{23, 26, 9}{4, 54, 63}{53, 68, 59}{77, 5, 64}{15, 2, 70}{68, 63, 8}{15, 44, 57}{23, 79, 33}{38, 25, 31}{61, 28, 68}{61, 6, 43}{53, 63, 18}{7, 76, 52}{56, 67, 38}{53, 73, 52}{28, 57, 72}{35, 20, 33}

{38, 64, 9}{68, 24, 12}{15, 6, 11}{34, 72, 35}{61, 25, 24}{55, 75, 45}{56, 50, 47}{20, 71, 2}{34, 10, 24}{65, 62, 15}{42, 16, 54}{68, 41, 30}{70, 73, 62}{6, 71, 74}{61, 21, 58}{70, 69, 58}{20, 62, 39}{23, 0, 76}{1, 10, 30}{72, 64, 53}{20, 8, 25}{35, 57, 66}{23, 25, 6}{10, 32, 56}{4, 30, 18}{79, 66, 68}{34, 15, 78}{47, 45, 36}{3, 5, 55}{33, 41, 61}{42, 79, 13}{33, 40, 45}{73, 55, 56}{29, 6, 7}{50, 13, 55}{31, 9, 1}{79, 36, 39}{5, 75, 33}{49, 28, 50}{19, 28, 78}{0, 37, 25}{57, 22, 65}{75, 52, 71}{71, 58, 1}{4, 73, 74}


{62, 60, 34}{4, 0, 15}{48, 33, 22}{53, 79, 14}{71, 38, 48}{9, 13, 32}{25, 70, 40}{26, 27, 53}{50, 32, 63}{54, 19, 44}{40, 22, 63}{56, 53, 71}{63, 12, 13}{63, 58, 9}{21, 32, 66}{29, 23, 78}{39, 8, 2}{59, 30, 34}{13, 10, 22}{62, 38, 19}{5, 0, 36}{50, 36, 10}{36, 41, 16}{50, 62, 57}{49, 38, 21}{37, 33, 64}{28, 77, 46}{56, 51, 15}{36, 26, 38}{24, 14, 28}{12, 67, 65}{24, 73, 78}{62, 33, 4}{38, 15, 43}{5, 67, 54}{54, 26, 21}{73, 61, 46}{31, 73, 7}{11, 14, 0}{53, 41, 29}{13, 4, 49}{6, 28, 65}{23, 70, 30}{21, 46, 6}{66, 60, 61}

{74, 72, 12}{0, 45, 31}{33, 32, 57}{0, 49, 57}{55, 77, 36}{45, 49, 58}{53, 6, 45}{37, 16, 1}{52, 50, 23}{31, 60, 69}{49, 70, 18}{57, 74, 13}{27, 1, 21}{32, 43, 60}{27, 72, 22}{20, 64, 14}{36, 33, 3}{66, 49, 27}{19, 9, 4}{4, 7, 61}{77, 74, 8}{44, 52, 72}{5, 19, 43}{52, 27, 9}{37, 76, 34}{29, 64, 69}{78, 61, 74}{12, 7, 27}{47, 29, 51}{59, 13, 71}{59, 46, 31}{46, 16, 40}{1, 70, 47}{47, 60, 19}{59, 29, 36}{56, 42, 28}{77, 38, 47}{11, 71, 21}{74, 22, 9}{6, 63, 36}{24, 11, 49}{18, 15, 12}{70, 0, 60}{18, 5, 23}{27, 20, 38}

{15, 14, 75}{7, 38, 51}{67, 31, 76}{6, 5, 52}{53, 77, 76}{37, 78, 63}{71, 79, 40}{73, 45, 23}{77, 18, 75}{15, 7, 41}{75, 57, 54}{37, 9, 40}{72, 69, 1}{60, 54, 2}{23, 34, 27}{28, 75, 10}{12, 46, 4}{42, 52, 18}{71, 73, 32}{58, 6, 48}{65, 4, 3}{45, 79, 9}{32, 47, 24}{51, 24, 41}{21, 63, 76}{42, 76, 9}{18, 19, 65}{41, 18, 31}{9, 53, 12}{16, 57, 27}{19, 17, 55}{31, 54, 71}{39, 10, 78}{47, 72, 7}{44, 74, 25}{5, 40, 39}{68, 32, 42}{58, 52, 47}{74, 31, 30}{9, 44, 16}{21, 77, 51}{75, 13, 25}{4, 76, 25}{32, 35, 22}{2, 27, 45}

{12, 19, 0}{53, 13, 16}{5, 26, 13}{69, 73, 0}{70, 68, 48}{35, 23, 65}{67, 9, 30}{60, 4, 56}{53, 78, 2}{42, 63, 2}{71, 57, 78}{47, 65, 78}{49, 75, 67}{24, 23, 59}{66, 1, 78}{51, 69, 2}{44, 78, 31}{30, 12, 5}{13, 15, 37}{31, 77, 66}{76, 15, 20}{70, 65, 79}{15, 59, 39}{46, 47, 18}{14, 74, 41}{35, 24, 53}{77, 62, 2}{56, 46, 54}{23, 58, 68}{24, 3, 21}{26, 43, 24}{52, 60, 21}{69, 15, 24}{59, 70, 55}{19, 26, 79}{71, 65, 45}{20, 24, 0}{43, 67, 71}{39, 30, 53}{43, 57, 40}{9, 70, 51}{30, 47, 73}{54, 29, 33}{35, 70, 52}{17, 32, 4}


{38, 5, 46}{38, 35, 8}{18, 79, 58}{0, 28, 47}{34, 67, 25}{9, 71, 61}{60, 58, 24}{60, 40, 42}{11, 39, 32}{51, 44, 18}{0, 75, 72}{32, 18, 25}{75, 73, 1}{48, 26, 4}{13, 66, 38}{44, 38, 75}{55, 0, 42}{7, 33, 34}{54, 68, 64}{57, 6, 26}{44, 47, 41}{8, 46, 71}{72, 60, 29}{26, 12, 50}{66, 63, 11}{41, 65, 66}{37, 29, 65}{11, 74, 38}{77, 3, 15}{25, 17, 53}{75, 63, 19}{40, 13, 47}{9, 62, 21}{12, 51, 42}{51, 40, 49}{60, 10, 27}{50, 78, 43}{75, 31, 32}{75, 66, 69}{69, 79, 35}{67, 53, 28}{27, 41, 55}{17, 16, 15}{13, 35, 18}{9, 50, 33}

{0, 52, 62}{42, 77, 1}{52, 74, 16}{59, 20, 6}{63, 69, 30}{19, 42, 57}{57, 60, 11}{43, 14, 76}{55, 1, 35}{30, 19, 29}{71, 60, 68}{50, 48, 7}{31, 27, 28}{78, 69, 22}{77, 20, 60}{38, 59, 37}{16, 73, 19}{79, 43, 54}{72, 31, 62}{16, 61, 79}{56, 78, 0}{52, 26, 14}{75, 35, 62}{4, 11, 29}{49, 61, 23}{79, 64, 59}{65, 44, 77}{77, 49, 43}{16, 49, 76}{15, 23, 19}{20, 13, 7}{20, 37, 57}{23, 64, 21}{56, 75, 23}{62, 24, 54}{58, 78, 27}{48, 24, 36}{65, 46, 13}{43, 12, 10}{11, 34, 68}{6, 30, 16}{27, 35, 5}{51, 1, 52}{64, 1, 6}{21, 43, 74}

{25, 30, 65}{14, 72, 73}{57, 58, 12}{10, 31, 21}{77, 17, 71}{50, 44, 53}{74, 69, 49}{46, 19, 32}{27, 63, 48}{1, 62, 13}{65, 54, 50}{72, 71, 36}{9, 5, 47}{5, 76, 22}{58, 20, 53}{72, 54, 76}{37, 72, 55}{34, 19, 6}{17, 29, 46}{20, 78, 40}{53, 38, 4}{79, 75, 24}{18, 29, 43}{39, 19, 68}{17, 39, 63}{61, 55, 51}{35, 58, 64}{79, 51, 10}{52, 43, 25}{60, 35, 25}{44, 20, 23}{65, 73, 8}{33, 71, 70}{11, 5, 70}{9, 24, 66}{28, 62, 36}{32, 65, 38}{1, 60, 38}{40, 50, 61}{38, 58, 55}{73, 26, 68}{74, 75, 68}{3, 62, 12}{21, 12, 17}{65, 63, 34}

{45, 46, 69}{68, 25, 15}{29, 40, 35}{34, 45, 26}{58, 16, 3}{26, 30, 60}{16, 66, 23}{66, 51, 17}{1, 56, 14}{23, 51, 72}{11, 67, 55}{72, 32, 58}{74, 50, 3}{21, 29, 75}{40, 7, 0}{75, 76, 78}{71, 27, 47}{63, 14, 23}{7, 5, 68}{29, 39, 14}{11, 20, 47}{49, 42, 7}{15, 53, 49}{74, 15, 55}{5, 42, 41}{79, 73, 37}{77, 39, 24}{20, 48, 30}{58, 51, 28}{64, 22, 49}{27, 70, 77}{15, 54, 8}{1, 48, 18}{8, 0, 34}{43, 22, 51}{8, 29, 12}{47, 25, 69}{46, 36, 70}{7, 60, 45}{37, 30, 45}{50, 30, 8}{61, 57, 76}{21, 34, 36}{3, 73, 11}{32, 69, 12}


{32, 1, 59}{52, 28, 69}{33, 44, 27}{1, 54, 0}{70, 21, 72}{37, 39, 43}{65, 2, 0}{18, 27, 69}{7, 46, 75}{7, 57, 18}{34, 61, 64}{24, 57, 30}{63, 0, 38}{54, 77, 35}{74, 35, 47}{78, 51, 5}{23, 36, 13}{26, 72, 15}{51, 27, 54}{44, 22, 46}{3, 39, 0}{33, 76, 74}{3, 2, 72}{74, 0, 51}{20, 18, 16}{72, 77, 79}{48, 67, 14}{30, 71, 49}{40, 58, 14}{35, 71, 41}{75, 2, 48}{30, 42, 75}{65, 36, 7}{60, 79, 48}{33, 51, 13}{70, 41, 76}{45, 39, 12}{77, 11, 9}{76, 29, 56}{13, 73, 48}{25, 78, 49}{8, 11, 78}{60, 14, 9}{6, 17, 67}{23, 12, 38}

{48, 52, 29}{52, 77, 67}{61, 14, 54}{7, 21, 14}{43, 34, 73}{74, 40, 36}{3, 31, 14}{17, 5, 62}{26, 37, 7}{69, 50, 39}{79, 67, 7}{77, 34, 40}{72, 41, 38}{0, 61, 67}{74, 53, 66}{61, 35, 26}{8, 33, 43}{15, 61, 36}{54, 18, 9}{21, 35, 59}{28, 39, 38}{9, 49, 59}{47, 42, 21}{8, 19, 52}{79, 17, 57}{14, 19, 37}{55, 65, 9}{60, 65, 53}{70, 31, 42}{28, 30, 22}{4, 21, 41}{50, 21, 19}{75, 3, 9}{66, 73, 5}{14, 42, 36}{21, 45, 68}{70, 45, 16}{39, 6, 47}{43, 30, 2}{72, 67, 20}{51, 25, 46}{26, 11, 22}{38, 18, 10}{79, 3, 52}{71, 10, 76}

{16, 59, 50}{46, 2, 9}{71, 22, 24}{53, 57, 46}{64, 75, 60}{56, 69, 6}{33, 72, 42}{5, 24, 31}{11, 50, 42}{64, 26, 46}{19, 36, 27}{66, 30, 56}{44, 1, 40}{62, 44, 69}{5, 15, 71}{66, 62, 48}{59, 61, 47}{38, 42, 34}{19, 66, 58}{37, 11, 12}{9, 29, 15}{77, 50, 37}{63, 3, 28}{69, 20, 55}{16, 34, 28}{16, 55, 26}{68, 35, 0}{50, 25, 64}{21, 0, 13}{2, 76, 26}{71, 44, 0}{13, 31, 34}{66, 12, 54}{66, 72, 25}{45, 19, 25}{20, 9, 10}{77, 25, 14}{20, 1, 22}{4, 47, 34}{57, 45, 1}{73, 2, 49}{67, 73, 60}{13, 72, 78}{1, 19, 76}{67, 40, 21}

{49, 62, 29}{52, 54, 45}{37, 28, 71}{25, 54, 3}{27, 40, 30}{13, 70, 44}{20, 70, 19}{71, 16, 4}{12, 77, 73}{20, 79, 34}{64, 27, 62}{36, 54, 30}{17, 73, 27}{18, 64, 78}{29, 57, 38}{24, 64, 76}{61, 44, 5}{47, 68, 67}{69, 77, 68}{29, 28, 26}{27, 46, 67}{77, 0, 22}{72, 16, 5}{45, 62, 10}{27, 50, 15}{70, 56, 12}{60, 18, 55}{26, 49, 3}{7, 78, 35}{20, 42, 73}{33, 19, 24}{67, 70, 8}{38, 3, 69}{6, 2, 41}{8, 5, 48}{37, 54, 41}{62, 40, 68}{47, 66, 43}{66, 22, 15}{72, 49, 19}{62, 79, 76}{11, 16, 25}{46, 49, 20}{58, 73, 59}{46, 15, 58}


{45, 63, 64}{16, 60, 8}{18, 3, 71}{12, 41, 78}{8, 22, 55}{25, 39, 21}{16, 62, 22}{28, 59, 5}{59, 60, 41}{74, 29, 2}{13, 2, 19}{69, 48, 61}{2, 61, 38}{78, 48, 3}{10, 70, 37}{66, 7, 3}{40, 2, 59}{66, 45, 59}{50, 45, 5}{14, 13, 69}{32, 3, 37}{49, 32, 36}{53, 54, 47}{69, 59, 54}{24, 29, 27}{32, 34, 52}{11, 44, 48}{59, 42, 67}{17, 52, 61}{43, 20, 41}{14, 51, 45}{51, 59, 65}{68, 38, 50}{2, 67, 33}{49, 12, 35}{28, 35, 9}{38, 24, 16}{58, 36, 25}{52, 64, 40}{18, 72, 39}{64, 19, 7}{24, 67, 63}{44, 66, 6}{31, 19, 11}{44, 39, 4}

{22, 39, 60}{74, 32, 28}{42, 17, 24}{16, 29, 31}{67, 44, 45}{8, 61, 31}{33, 18, 26}{59, 48, 12}{60, 13, 6}{51, 73, 36}{48, 35, 39}{29, 0, 66}{3, 60, 46}{25, 55, 63}{20, 66, 26}{12, 40, 55}{18, 74, 67}{74, 60, 5}{0, 43, 9}{48, 65, 40}{4, 55, 79}{22, 59, 52}{55, 57, 31}{78, 36, 60}{24, 55, 6}{23, 48, 28}{41, 56, 62}{54, 49, 55}{0, 6, 33}{26, 40, 31}{13, 54, 39}{57, 8, 36}{7, 22, 25}{11, 7, 54}{6, 49, 10}{68, 56, 27}{4, 28, 2}{42, 8, 6}{9, 7, 17}{63, 52, 57}{49, 41, 52}{58, 56, 44}{47, 8, 64}{24, 1, 4}{14, 49, 34}

{71, 51, 34}{70, 3, 53}{26, 56, 25}{74, 65, 24}{55, 66, 14}{72, 61, 30}{66, 76, 46}{69, 11, 17}{50, 20, 31}{19, 41, 10}{65, 31, 64}{12, 61, 20}{49, 8, 79}{52, 56, 33}{32, 23, 54}{65, 76, 58}{52, 2, 24}{27, 32, 14}{28, 8, 7}{66, 64, 4}{64, 12, 71}{25, 33, 59}{47, 76, 48}{49, 60, 63}{52, 65, 10}{30, 15, 52}{39, 1, 61}{21, 28, 55}{12, 33, 47}{58, 8, 13}{75, 53, 34}{1, 29, 25}{26, 63, 71}{29, 32, 79}{43, 13, 56}{45, 22, 3}{69, 9, 8}{14, 6, 68}{34, 70, 57}{44, 17, 2}{17, 35, 10}{57, 51, 48}{2, 14, 10}{37, 44, 24}{27, 74, 39}

{56, 63, 35}{6, 75, 50}{27, 13, 76}{39, 46, 33}{4, 43, 45}{43, 16, 35}{76, 59, 3}{73, 28, 54}{35, 45, 15}{21, 48, 15}{42, 71, 66}{28, 66, 33}{1, 79, 2}{75, 39, 70}{33, 21, 30}{37, 66, 8}{32, 62, 26}{40, 18, 28}{34, 56, 74}{4, 69, 42}{17, 31, 48}{78, 6, 77}{18, 62, 11}{4, 27, 8}{44, 49, 68}{23, 17, 41}{39, 16, 67}{61, 27, 65}{40, 73, 15}{71, 19, 69}{79, 44, 21}{61, 37, 42}{46, 52, 55}{19, 56, 61}{79, 38, 30}{2, 56, 64}{78, 4, 59}{36, 35, 76}{42, 3, 27}{68, 37, 2}{2, 31, 36}{48, 49, 56}{10, 3, 61}{58, 7, 30}{65, 11, 72}


{1, 36, 11}{40, 75, 4}{4, 5, 57}{69, 41, 34}{72, 9, 6}{8, 26, 44}{10, 67, 15}{4, 14, 70}{67, 62, 37}{11, 64, 28}{55, 29, 68}{47, 62, 55}{28, 13, 41}{17, 58, 75}{53, 0, 10}{33, 55, 78}{50, 51, 60}{33, 77, 63}{74, 48, 37}{32, 8, 45}{18, 0, 17}{1, 28, 15}

{57, 21, 2}{49, 47, 37}{68, 1, 46}{13, 30, 17}{8, 59, 62}{18, 56, 59}{70, 32, 61}{19, 59, 74}{7, 63, 43}{36, 56, 9}{63, 5, 1}{40, 53, 19}{40, 10, 66}{0, 27, 79}{54, 58, 34}{74, 17, 45}{4, 67, 23}{38, 45, 76}{14, 18, 22}{9, 78, 68}{65, 14, 5}{70, 78, 26}

{41, 75, 8}{78, 67, 32}{78, 17, 54}{14, 50, 71}{22, 21, 73}{32, 76, 6}{1, 53, 8}{50, 29, 70}{46, 41, 0}{23, 22, 31}{32, 7, 44}{1, 41, 50}{16, 75, 47}{70, 43, 64}{43, 44, 42}{53, 48, 55}{30, 32, 77}{61, 53, 11}{59, 26, 0}{50, 24, 46}{37, 22, 56}{40, 3, 6}

{74, 23, 62}{42, 25, 48}{36, 43, 69}{10, 54, 48}{3, 13, 68}{64, 3, 41}{68, 16, 51}{66, 39, 52}{46, 35, 42}{65, 43, 68}{28, 45, 20}{6, 51, 62}{27, 6, 37}{65, 21, 56}{44, 63, 59}{47, 14, 57}{74, 1, 7}{53, 31, 43}{1, 34, 3}{63, 70, 74}


g = 5, u = 20:

{10, 34, 64}{51, 13, 87}{95, 11, 58}{75, 79, 11}{55, 96, 21}{8, 49, 63}{44, 98, 23}{72, 27, 60}{86, 4, 13}{68, 53, 20}{45, 14, 4}{5, 46, 91}{23, 10, 82}{13, 9, 35}{63, 21, 25}{2, 92, 67}{26, 89, 17}{49, 72, 39}{98, 40, 10}{36, 72, 61}{48, 36, 4}{9, 27, 93}{10, 18, 81}{21, 51, 23}{91, 8, 53}{84, 47, 39}{12, 54, 20}{19, 62, 1}{51, 34, 16}{31, 54, 82}{52, 11, 19}{93, 88, 67}{40, 36, 45}{99, 78, 71}{64, 81, 13}{14, 58, 6}{99, 58, 4}{38, 19, 7}{50, 53, 40}{52, 95, 94}{62, 56, 64}{7, 34, 18}{67, 8, 69}{91, 27, 37}{33, 44, 17}

{79, 76, 74}{82, 84, 40}{0, 62, 85}{65, 92, 41}{75, 98, 14}{83, 62, 14}{28, 96, 69}{6, 94, 69}{50, 8, 31}{87, 64, 37}{53, 48, 78}{47, 8, 89}{37, 46, 95}{49, 25, 71}{96, 32, 83}{39, 11, 29}{0, 65, 78}{34, 26, 91}{60, 89, 85}{26, 4, 68}{4, 0, 95}{58, 25, 42}{82, 25, 32}{71, 26, 84}{1, 10, 94}{71, 93, 21}{32, 0, 86}{98, 1, 71}{18, 27, 49}{16, 46, 67}{35, 3, 11}{65, 12, 59}{69, 12, 19}{8, 18, 40}{74, 83, 0}{93, 72, 46}{66, 24, 47}{25, 2, 59}{23, 55, 62}{37, 13, 61}{20, 38, 1}{3, 99, 14}{83, 28, 20}{10, 38, 71}{2, 6, 53}

{0, 70, 94}{43, 34, 4}{98, 74, 37}{14, 10, 67}{3, 15, 89}{31, 59, 73}{0, 37, 10}{38, 43, 44}{73, 66, 98}{77, 86, 21}{91, 6, 77}{35, 20, 37}{79, 42, 63}{29, 1, 36}{34, 73, 17}{42, 43, 98}{35, 87, 79}{0, 55, 5}{53, 47, 28}{55, 40, 29}{66, 77, 52}{37, 99, 22}{86, 15, 56}{30, 48, 74}{89, 11, 96}{16, 74, 29}{4, 82, 19}{53, 15, 66}{90, 98, 4}{0, 25, 36}{28, 99, 17}{32, 31, 55}{36, 97, 43}{45, 64, 59}{36, 11, 85}{42, 8, 32}{13, 15, 68}{12, 48, 49}{21, 45, 88}{0, 81, 12}{3, 13, 60}{47, 20, 78}{15, 31, 85}{4, 60, 94}{21, 70, 92}

{79, 0, 23}{60, 91, 67}{93, 7, 62}{46, 44, 21}{76, 37, 51}{49, 94, 42}{60, 37, 55}{1, 25, 44}{32, 59, 71}{11, 22, 48}{15, 29, 77}{13, 80, 59}{82, 5, 60}{26, 72, 99}{49, 83, 38}{47, 45, 38}{67, 34, 36}{4, 46, 62}{67, 37, 75}{66, 65, 51}{14, 16, 71}{73, 70, 43}{15, 11, 62}{8, 51, 93}{12, 79, 7}{89, 81, 38}{51, 88, 24}{55, 49, 77}{46, 38, 15}{17, 15, 39}{52, 90, 20}{79, 46, 36}{1, 63, 92}{54, 87, 58}{76, 21, 42}{29, 7, 48}{30, 63, 68}{26, 41, 70}{17, 21, 11}{61, 22, 79}{78, 52, 34}{7, 40, 97}{59, 69, 18}{31, 34, 70}{91, 40, 92}


{59, 3, 97}{18, 32, 20}{42, 17, 67}{97, 16, 6}{52, 25, 9}{99, 89, 44}{40, 76, 90}{67, 20, 15}{85, 76, 72}{91, 48, 52}{11, 87, 41}{97, 5, 90}{61, 68, 94}{50, 5, 61}{63, 10, 33}{76, 6, 83}{38, 21, 4}{69, 73, 71}{0, 14, 61}{72, 88, 34}{96, 22, 20}{24, 15, 28}{61, 3, 53}{49, 23, 88}{65, 4, 80}{28, 77, 78}{86, 44, 54}{93, 61, 47}{93, 23, 86}{43, 60, 96}{96, 10, 59}{4, 16, 69}{68, 39, 38}{49, 47, 68}{31, 92, 60}{69, 78, 24}{73, 95, 49}{74, 2, 84}{94, 75, 91}{3, 55, 74}{86, 75, 78}{82, 91, 9}{79, 20, 91}{22, 87, 16}{66, 67, 68}

{94, 46, 45}{68, 76, 86}{84, 17, 94}{49, 79, 45}{45, 78, 9}{17, 36, 20}{82, 21, 43}{57, 99, 38}{36, 31, 6}{76, 3, 33}{21, 27, 99}{92, 27, 23}{35, 61, 58}{98, 63, 28}{72, 0, 51}{17, 12, 74}{83, 89, 71}{1, 17, 30}{34, 53, 37}{93, 37, 68}{3, 47, 41}{97, 18, 47}{95, 83, 18}{66, 40, 63}{70, 7, 96}{62, 21, 67}{35, 33, 28}{21, 5, 84}{27, 34, 95}{12, 58, 77}{36, 70, 24}{85, 52, 69}{19, 17, 49}{44, 18, 76}{92, 94, 66}{70, 79, 69}{49, 65, 37}{91, 84, 15}{22, 7, 15}{71, 65, 20}{72, 38, 55}{71, 35, 12}{0, 97, 1}{81, 86, 55}{44, 2, 9}

{74, 18, 60}{80, 17, 68}{22, 95, 81}{29, 20, 61}{56, 12, 63}{42, 19, 6}{58, 23, 57}{21, 53, 12}{2, 79, 71}{15, 37, 94}{24, 67, 83}{30, 4, 35}{32, 69, 33}{7, 32, 50}{2, 10, 57}{92, 77, 99}{79, 15, 52}{12, 90, 46}{96, 62, 9}{21, 15, 9}{82, 94, 39}{7, 13, 74}{72, 77, 14}{10, 54, 99}{93, 87, 43}{69, 84, 46}{11, 46, 70}{70, 39, 32}{70, 67, 19}{72, 18, 84}{19, 95, 36}{61, 49, 60}{53, 70, 44}{1, 35, 64}{46, 58, 32}{98, 93, 25}{2, 20, 39}{50, 69, 42}{14, 22, 86}{48, 64, 21}{74, 95, 23}{35, 22, 80}{56, 20, 51}{21, 91, 7}{17, 7, 66}

{23, 35, 70}{46, 14, 60}{51, 86, 29}{75, 0, 27}{59, 94, 62}{18, 68, 92}{4, 79, 55}{90, 54, 77}{42, 55, 18}{25, 61, 48}{61, 86, 8}{57, 13, 95}{44, 34, 63}{24, 90, 85}{28, 1, 3}{47, 13, 10}{93, 10, 19}{36, 44, 49}{97, 70, 52}{31, 89, 87}{25, 87, 17}{7, 81, 25}{62, 91, 10}{32, 27, 68}{74, 44, 81}{72, 81, 40}{14, 52, 80}{28, 22, 84}{39, 16, 9}{73, 54, 60}{41, 35, 86}{5, 81, 47}{86, 10, 5}{81, 82, 30}{61, 27, 71}{48, 65, 94}{36, 12, 68}{62, 84, 49}{6, 41, 15}{4, 87, 96}{47, 59, 30}{21, 6, 18}{86, 79, 88}{95, 77, 33}{2, 43, 95}


{22, 94, 72}{8, 15, 74}{82, 45, 0}{30, 69, 64}{83, 52, 84}{83, 94, 78}{25, 24, 11}{82, 63, 48}{50, 37, 59}{62, 28, 75}{96, 86, 24}{33, 57, 25}{15, 42, 60}{81, 20, 26}{45, 18, 2}{78, 95, 39}{44, 47, 19}{9, 58, 8}{52, 26, 37}{18, 33, 12}{59, 54, 7}{43, 46, 51}{8, 73, 11}{66, 72, 43}{71, 23, 45}{75, 69, 20}{52, 18, 50}{38, 50, 62}{5, 6, 62}{6, 20, 30}{52, 65, 13}{30, 33, 15}{28, 23, 94}{64, 61, 99}{33, 98, 91}{83, 27, 42}{20, 99, 43}{43, 5, 54}{20, 86, 94}{92, 98, 89}{57, 43, 88}{15, 0, 47}{87, 20, 76}{16, 24, 40}{11, 66, 60}

{6, 87, 29}{41, 67, 98}{46, 10, 20}{37, 72, 11}{16, 55, 2}{28, 30, 43}{75, 68, 84}{20, 49, 14}{7, 60, 6}{14, 57, 68}{88, 5, 58}{67, 30, 73}{91, 89, 78}{61, 26, 23}{89, 46, 23}{4, 1, 72}{97, 31, 56}{37, 66, 71}{31, 20, 72}{67, 89, 77}{76, 65, 9}{0, 68, 52}{92, 90, 56}{27, 41, 13}{54, 96, 15}{10, 83, 44}{11, 67, 28}{8, 6, 1}{43, 62, 68}{1, 74, 58}{93, 11, 6}{80, 6, 12}{74, 89, 50}{9, 4, 67}{14, 66, 97}{23, 29, 59}{34, 57, 82}{48, 93, 77}{40, 17, 9}{19, 55, 90}{84, 86, 48}{68, 60, 33}{32, 30, 75}{19, 21, 32}{99, 50, 15}

{99, 55, 69}{72, 74, 21}{66, 23, 69}{58, 67, 43}{21, 83, 69}{75, 74, 71}{10, 6, 65}{41, 32, 60}{31, 98, 0}{33, 66, 8}{85, 34, 97}{56, 40, 79}{89, 94, 58}{57, 67, 55}{91, 42, 56}{25, 91, 18}{33, 54, 19}{44, 82, 71}{74, 97, 11}{67, 80, 64}{50, 84, 56}{78, 70, 66}{76, 73, 23}{62, 24, 31}{65, 84, 61}{73, 80, 44}{29, 38, 52}{37, 70, 88}{76, 25, 34}{11, 44, 32}{46, 80, 78}{46, 42, 48}{59, 86, 98}{16, 61, 70}{66, 3, 25}{4, 6, 78}{33, 84, 70}{34, 49, 0}{84, 96, 81}{36, 13, 66}{97, 41, 72}{21, 89, 22}{49, 41, 78}{3, 21, 16}{81, 99, 94}

{62, 97, 53}{46, 98, 65}{8, 78, 76}{97, 79, 73}{83, 35, 57}{61, 12, 24}{48, 58, 72}{39, 33, 4}{29, 13, 32}{53, 19, 60}{75, 3, 40}{30, 8, 44}{49, 21, 75}{39, 0, 93}{32, 66, 49}{68, 73, 21}{17, 3, 79}{54, 91, 63}{56, 95, 48}{73, 15, 92}{80, 58, 84}{38, 3, 64}{43, 10, 32}{39, 53, 89}{91, 13, 30}{88, 69, 0}{97, 96, 71}{3, 57, 86}{64, 16, 32}{23, 37, 39}{27, 54, 28}{13, 17, 24}{41, 29, 84}{16, 77, 82}{73, 27, 24}{68, 46, 96}{99, 6, 73}{30, 54, 80}{10, 17, 72}{95, 93, 91}{59, 84, 67}{78, 15, 32}{26, 67, 32}{40, 59, 70}{34, 6, 3}


{62, 37, 90}{15, 1, 65}{87, 90, 45}{23, 15, 64}{91, 43, 47}{82, 17, 58}{63, 81, 70}{27, 86, 30}{92, 20, 13}{8, 5, 37}{74, 10, 52}{71, 50, 19}{26, 15, 2}{53, 16, 83}{89, 76, 62}{30, 24, 42}{49, 4, 93}{11, 27, 10}{54, 3, 37}{11, 83, 65}{20, 5, 27}{23, 12, 38}{8, 27, 22}{47, 73, 35}{96, 88, 1}{98, 21, 39}{24, 46, 29}{57, 85, 9}{71, 28, 58}{60, 90, 9}{29, 50, 2}{41, 58, 34}{29, 19, 73}{41, 40, 83}{45, 26, 48}{91, 41, 69}{42, 65, 40}{27, 89, 84}{48, 39, 13}{40, 28, 34}{55, 1, 78}{78, 27, 85}{21, 65, 36}{13, 90, 25}{79, 77, 50}

{12, 73, 85}{30, 16, 26}{72, 35, 59}{40, 2, 46}{40, 44, 62}{78, 96, 19}{50, 41, 33}{80, 39, 85}{20, 19, 84}{57, 65, 32}{32, 51, 6}{35, 46, 63}{90, 31, 14}{29, 99, 8}{85, 71, 13}{67, 72, 96}{66, 22, 1}{59, 81, 36}{92, 96, 6}{37, 25, 29}{40, 68, 95}{29, 97, 68}{75, 10, 29}{20, 95, 7}{58, 69, 68}{14, 5, 76}{52, 35, 7}{25, 67, 53}{11, 7, 23}{94, 85, 32}{23, 90, 8}{22, 39, 91}{32, 45, 17}{77, 84, 98}{70, 57, 45}{44, 78, 7}{48, 3, 87}{22, 36, 74}{12, 66, 89}{32, 84, 76}{45, 95, 62}{75, 45, 76}{88, 56, 4}{35, 88, 99}{80, 50, 87}

{29, 53, 45}{18, 99, 51}{56, 21, 87}{56, 19, 22}{56, 35, 68}{87, 53, 65}{56, 37, 2}{91, 87, 83}{27, 80, 1}{65, 28, 97}{77, 42, 31}{19, 57, 28}{42, 3, 12}{18, 48, 9}{73, 75, 82}{57, 36, 60}{16, 86, 52}{89, 16, 25}{42, 16, 10}{6, 72, 82}{47, 40, 69}{83, 51, 85}{84, 60, 79}{16, 44, 12}{64, 50, 92}{76, 2, 61}{78, 22, 31}{43, 81, 85}{83, 15, 58}{94, 2, 11}{29, 67, 3}{9, 77, 32}{99, 76, 46}{49, 50, 96}{42, 34, 66}{32, 37, 28}{8, 14, 84}{95, 89, 14}{34, 5, 38}{33, 11, 5}{96, 94, 57}{1, 50, 57}{39, 28, 86}{81, 56, 49}{86, 19, 34}

{88, 98, 11}{18, 29, 14}{19, 37, 89}{33, 87, 2}{39, 27, 35}{40, 32, 61}{47, 2, 72}{14, 85, 92}{58, 50, 81}{77, 8, 87}{63, 22, 71}{8, 96, 85}{58, 24, 39}{81, 57, 39}{30, 89, 52}{4, 53, 31}{79, 34, 80}{36, 5, 87}{75, 23, 56}{56, 70, 29}{2, 7, 0}{68, 82, 3}{57, 56, 89}{51, 39, 40}{14, 43, 15}{36, 33, 99}{52, 46, 59}{7, 24, 55}{33, 82, 67}{87, 1, 69}{92, 4, 5}{29, 54, 71}{25, 73, 94}{31, 74, 88}{38, 40, 48}{43, 31, 12}{71, 7, 46}{71, 18, 80}{78, 82, 61}{29, 12, 78}{97, 63, 55}{41, 48, 51}{54, 89, 24}{31, 65, 26}{91, 29, 64}


{96, 37, 44}{33, 46, 49}{85, 47, 46}{58, 31, 45}{67, 61, 38}{41, 71, 64}{51, 78, 10}{52, 42, 73}{86, 7, 92}{51, 62, 35}{74, 78, 68}{1, 39, 7}{97, 54, 2}{23, 22, 41}{18, 31, 67}{14, 51, 33}{84, 66, 99}{94, 50, 9}{4, 12, 28}{80, 10, 41}{85, 82, 59}{20, 59, 55}{30, 92, 25}{60, 64, 97}{89, 93, 40}{7, 45, 72}{65, 69, 3}{7, 83, 8}{90, 64, 43}{81, 19, 16}{56, 41, 14}{78, 56, 33}{82, 24, 20}{85, 42, 93}{25, 8, 79}{3, 90, 80}{50, 75, 22}{63, 5, 39}{56, 6, 17}{49, 87, 15}{4, 11, 42}{50, 0, 24}{53, 98, 22}{41, 36, 93}{26, 42, 14}

{24, 57, 18}{13, 62, 88}{8, 52, 3}{98, 47, 32}{20, 4, 23}{25, 46, 74}{15, 44, 90}{72, 54, 75}{97, 69, 86}{34, 96, 23}{18, 77, 23}{11, 99, 63}{9, 33, 20}{20, 64, 89}{27, 65, 77}{59, 28, 49}{5, 3, 32}{6, 38, 35}{28, 72, 29}{35, 74, 65}{74, 93, 28}{7, 80, 43}{42, 0, 35}{58, 19, 63}{64, 63, 75}{33, 89, 61}{82, 35, 53}{45, 77, 63}{31, 84, 37}{73, 0, 22}{53, 10, 85}{62, 79, 41}{45, 92, 51}{80, 9, 11}{86, 71, 70}{64, 98, 7}{25, 88, 15}{65, 99, 30}{41, 17, 59}{62, 86, 74}{65, 39, 73}{35, 36, 32}{61, 42, 95}{69, 44, 35}{90, 73, 78}

{52, 27, 17}{34, 13, 2}{84, 10, 25}{55, 68, 50}{59, 6, 48}{31, 28, 52}{49, 97, 51}{26, 55, 39}{72, 42, 80}{96, 58, 64}{11, 1, 40}{35, 45, 43}{56, 60, 59}{18, 39, 88}{99, 74, 56}{20, 48, 97}{37, 85, 41}{48, 75, 17}{76, 77, 41}{45, 27, 66}{91, 35, 16}{87, 40, 73}{60, 81, 17}{67, 50, 39}{63, 14, 9}{12, 11, 55}{50, 78, 60}{64, 52, 36}{53, 64, 72}{83, 82, 56}{82, 66, 55}{76, 80, 97}{80, 70, 49}{94, 24, 19}{81, 42, 68}{84, 16, 0}{45, 28, 10}{82, 15, 18}{63, 0, 87}{70, 1, 5}{8, 75, 57}{3, 18, 62}{39, 41, 96}{25, 47, 60}{9, 36, 98}

{81, 23, 31}{29, 96, 93}{0, 44, 77}{74, 77, 59}{0, 18, 64}{42, 45, 86}{4, 85, 70}{94, 98, 97}{30, 83, 36}{14, 96, 82}{63, 89, 59}{66, 50, 91}{96, 75, 66}{55, 92, 44}{5, 17, 98}{59, 66, 90}{68, 34, 59}{9, 0, 99}{47, 42, 36}{13, 97, 89}{63, 26, 62}{51, 69, 27}{52, 63, 96}{2, 88, 73}{24, 72, 71}{88, 41, 54}{78, 81, 67}{77, 56, 5}{98, 95, 69}{51, 50, 54}{61, 10, 56}{27, 38, 76}{5, 73, 18}{36, 27, 2}{36, 23, 84}{22, 9, 92}{93, 97, 84}{59, 42, 51}{13, 5, 94}{53, 80, 99}{45, 91, 44}{53, 17, 51}{72, 79, 33}{0, 30, 53}{88, 3, 27}


{19, 45, 83}{90, 99, 48}{9, 46, 54}{62, 47, 12}{90, 69, 34}{82, 80, 74}{22, 55, 47}{52, 23, 99}{13, 40, 99}{43, 40, 25}{72, 15, 70}{98, 80, 16}{82, 50, 65}{36, 71, 94}{88, 50, 36}{19, 80, 77}{51, 60, 75}{78, 63, 93}{85, 33, 55}{88, 22, 52}{68, 77, 2}{71, 77, 40}{15, 63, 76}{87, 86, 38}{1, 32, 2}{91, 86, 73}{47, 71, 88}{98, 62, 72}{79, 37, 9}{77, 25, 69}{53, 18, 75}{79, 10, 48}{22, 32, 34}{8, 80, 24}{54, 76, 26}{70, 18, 13}{80, 55, 45}{18, 96, 90}{79, 24, 14}{45, 15, 12}{2, 90, 28}{80, 81, 29}{63, 80, 61}{48, 70, 83}{18, 66, 61}

{14, 28, 50}{91, 49, 1}{70, 38, 17}{8, 39, 97}{9, 81, 6}{58, 0, 56}{94, 88, 29}{91, 28, 85}{71, 52, 55}{84, 1, 85}{85, 21, 58}{63, 41, 95}{35, 67, 48}{95, 64, 25}{79, 57, 21}{22, 54, 6}{61, 39, 30}{53, 86, 1}{69, 5, 22}{75, 58, 36}{23, 91, 80}{44, 13, 28}{45, 60, 52}{30, 41, 94}{98, 49, 35}{38, 80, 37}{32, 63, 90}{57, 30, 98}{96, 31, 2}{38, 69, 2}{96, 98, 45}{98, 2, 83}{30, 95, 71}{49, 90, 11}{90, 21, 95}{33, 1, 23}{4, 29, 57}{49, 85, 64}{50, 95, 17}{31, 5, 66}{91, 88, 65}{64, 55, 17}{55, 89, 10}{64, 82, 93}{96, 27, 53}

{76, 24, 43}{65, 54, 67}{75, 25, 80}{18, 93, 94}{89, 35, 5}{43, 50, 16}{13, 38, 96}{19, 13, 46}{79, 92, 28}{20, 3, 50}{75, 89, 4}{72, 30, 5}{8, 19, 92}{12, 9, 86}{75, 92, 81}{61, 51, 52}{14, 70, 27}{16, 17, 31}{48, 14, 37}{77, 11, 53}{93, 60, 69}{94, 47, 90}{36, 78, 3}{20, 58, 93}{22, 76, 13}{12, 22, 70}{54, 38, 92}{90, 58, 79}{11, 50, 86}{20, 34, 21}{87, 39, 75}{36, 14, 91}{56, 54, 85}{57, 84, 42}{9, 31, 30}{17, 78, 88}{45, 1, 68}{33, 27, 16}{92, 49, 82}{19, 87, 14}{76, 35, 60}{90, 74, 61}{84, 55, 30}{21, 40, 37}{68, 16, 11}

{1, 73, 14}{83, 90, 39}{16, 8, 94}{30, 37, 12}{80, 92, 95}{25, 12, 99}{10, 77, 73}{99, 75, 47}{62, 34, 92}{7, 37, 42}{93, 57, 22}{61, 57, 44}{51, 4, 2}{90, 22, 67}{86, 99, 60}{6, 45, 89}{62, 60, 16}{10, 95, 87}{60, 98, 48}{23, 54, 40}{62, 65, 8}{86, 65, 47}{19, 2, 41}{54, 83, 55}{11, 26, 56}{83, 46, 50}{61, 11, 34}{5, 49, 16}{42, 70, 89}{96, 99, 42}{96, 73, 48}{99, 16, 85}{9, 23, 53}{75, 1, 52}{84, 3, 73}{79, 64, 78}{27, 97, 19}{45, 74, 24}{76, 93, 1}{24, 92, 87}{93, 17, 83}{7, 89, 36}{73, 9, 7}{64, 77, 70}{87, 70, 74}


{10, 60, 8}{54, 98, 81}{26, 12, 96}{19, 5, 74}{40, 4, 27}{20, 25, 62}{30, 87, 85}{71, 0, 67}{46, 57, 92}{74, 63, 4}{92, 39, 36}{59, 76, 58}{19, 31, 3}{83, 59, 9}{37, 16, 63}{30, 45, 93}{57, 71, 62}{10, 68, 9}{44, 65, 72}{92, 16, 48}{41, 16, 18}{29, 98, 76}{61, 7, 85}{26, 0, 13}{85, 79, 18}{56, 71, 39}{83, 72, 13}{30, 14, 40}{32, 89, 48}{12, 51, 67}{79, 47, 51}{79, 54, 93}{17, 71, 4}{44, 51, 22}{30, 51, 77}{87, 81, 34}{32, 4, 97}{59, 15, 16}{36, 90, 53}{17, 63, 2}{81, 3, 4}{43, 79, 89}{26, 51, 94}{12, 57, 91}{1, 34, 46}

{2, 58, 60}{38, 11, 82}{51, 19, 9}{64, 54, 11}{55, 43, 27}{38, 73, 36}{65, 33, 96}{92, 11, 59}{38, 65, 79}{69, 82, 13}{11, 20, 57}{88, 59, 53}{34, 35, 8}{70, 47, 6}{6, 61, 75}{90, 89, 1}{22, 46, 64}{59, 78, 5}{53, 76, 57}{47, 96, 95}{90, 17, 65}{49, 31, 10}{99, 34, 83}{99, 5, 7}{54, 49, 53}{27, 94, 44}{19, 76, 88}{94, 80, 21}{24, 59, 93}{57, 49, 52}{34, 77, 47}{26, 98, 50}{54, 66, 0}{68, 22, 83}{26, 22, 18}{39, 42, 64}{88, 64, 40}{33, 97, 26}{39, 31, 76}{9, 47, 74}{86, 18, 17}{32, 99, 93}{28, 64, 5}{31, 35, 40}{0, 57, 41}

{53, 42, 5}{8, 36, 54}{0, 3, 46}{64, 73, 51}{0, 21, 8}{35, 92, 78}{82, 52, 87}{74, 42, 41}{30, 21, 97}{1, 56, 47}{44, 52, 56}{63, 73, 72}{55, 36, 51}{55, 70, 93}{84, 92, 53}{0, 38, 33}{13, 11, 14}{1, 37, 83}{24, 2, 35}{7, 10, 3}{93, 38, 90}{4, 54, 47}{21, 35, 14}{55, 34, 65}{21, 59, 26}{95, 65, 60}{24, 33, 52}{14, 47, 64}{94, 40, 12}{75, 77, 13}{98, 19, 61}{87, 66, 57}{65, 63, 29}{81, 76, 66}{52, 81, 62}{28, 76, 70}{8, 13, 56}{79, 96, 30}{48, 57, 5}{5, 52, 2}{48, 80, 47}{87, 12, 97}{81, 77, 24}{76, 11, 0}{45, 11, 84}

{37, 92, 58}{16, 93, 65}{9, 72, 87}{91, 74, 96}{88, 6, 55}{45, 67, 56}{91, 70, 2}{87, 71, 42}{41, 44, 31}{48, 66, 44}{58, 44, 29}{47, 21, 50}{51, 63, 84}{54, 95, 32}{11, 30, 18}{78, 42, 54}{47, 58, 26}{92, 3, 93}{78, 26, 40}{52, 53, 58}{39, 54, 25}{98, 85, 20}{54, 61, 45}{18, 4, 37}{81, 91, 90}{80, 66, 2}{23, 19, 68}{24, 10, 21}{97, 95, 9}{41, 68, 7}{2, 3, 30}{61, 87, 59}{35, 10, 97}{56, 72, 69}{20, 45, 16}{14, 55, 53}{13, 21, 31}{19, 64, 26}{77, 60, 83}{47, 31, 83}{8, 20, 70}{27, 15, 98}{35, 77, 26}{38, 97, 24}{90, 57, 7}


{76, 17, 91}{75, 41, 46}{66, 88, 95}{98, 70, 3}{48, 24, 23}{87, 46, 18}{76, 69, 7}{17, 46, 8}{93, 31, 75}{22, 60, 29}{62, 77, 36}{33, 92, 83}{18, 19, 35}{79, 53, 94}{24, 1, 26}{44, 97, 88}{67, 94, 63}{36, 18, 28}{24, 91, 32}{34, 48, 50}{15, 48, 71}{65, 70, 58}{14, 38, 32}{25, 55, 41}{99, 82, 70}{13, 58, 55}{75, 38, 85}{73, 28, 61}{88, 81, 14}{86, 33, 64}{23, 17, 85}{42, 88, 75}{62, 48, 27}{84, 88, 38}{90, 16, 75}{22, 49, 24}{81, 45, 8}{87, 68, 44}{56, 7, 30}{82, 1, 51}{4, 41, 52}{12, 34, 39}{77, 96, 61}{72, 90, 86}{90, 6, 0}

{92, 26, 10}{29, 21, 33}{89, 68, 72}{76, 52, 47}{71, 68, 90}{82, 37, 36}{74, 69, 26}{69, 14, 17}{90, 82, 41}{31, 29, 27}{24, 6, 37}{35, 93, 81}{43, 92, 74}{50, 73, 45}{16, 79, 1}{48, 85, 2}{31, 38, 63}{58, 97, 91}{61, 55, 91}{51, 80, 89}{59, 0, 43}{41, 20, 66}{74, 51, 57}{5, 71, 9}{14, 65, 7}{77, 1, 43}{61, 92, 69}{8, 55, 98}{84, 12, 95}{60, 26, 38}{57, 47, 63}{56, 53, 46}{82, 46, 97}{81, 27, 46}{68, 65, 64}{25, 51, 38}{58, 86, 40}{39, 60, 44}{56, 93, 66}{95, 6, 28}{75, 43, 26}{99, 95, 31}{42, 90, 29}{29, 92, 0}{49, 3, 58}

{86, 2, 89}{62, 30, 78}{63, 50, 13}{44, 67, 79}{31, 68, 79}{10, 12, 88}{23, 60, 30}{8, 43, 71}{7, 51, 58}{26, 29, 79}{60, 88, 87}{42, 9, 1}{61, 43, 9}{57, 78, 72}{80, 28, 56}{89, 65, 18}{84, 78, 87}{17, 96, 0}{30, 88, 46}{50, 35, 85}{31, 69, 57}{96, 51, 3}{31, 7, 94}{82, 89, 88}{5, 83, 29}{2, 12, 75}{27, 50, 12}{16, 13, 78}{14, 59, 44}{26, 25, 83}{5, 68, 24}{36, 80, 26}{77, 39, 46}{16, 7, 88}{64, 6, 57}{64, 74, 27}{10, 15, 36}{18, 54, 1}{75, 34, 24}{95, 85, 29}{89, 73, 41}{75, 70, 9}{73, 37, 81}{71, 76, 92}{35, 17, 54}

{2, 65, 81}{6, 98, 52}{10, 69, 39}{16, 73, 58}{62, 73, 32}{33, 74, 40}{49, 67, 86}{63, 7, 53}{3, 44, 95}{61, 62, 17}{19, 65, 75}{35, 25, 96}{26, 88, 9}{45, 39, 3}{74, 85, 67}{8, 38, 41}{81, 71, 53}{91, 38, 59}{26, 27, 90}{62, 70, 54}{97, 22, 45}{70, 25, 68}{54, 48, 69}{7, 4, 77}{0, 28, 89}{8, 4, 59}{62, 66, 39}{49, 2, 99}{95, 1, 59}{34, 45, 33}{4, 83, 73}{26, 8, 82}{65, 43, 22}{65, 24, 56}{69, 80, 62}{63, 6, 27}{24, 3, 9}{25, 86, 31}{88, 20, 63}{81, 48, 33}{3, 94, 56}{4, 25, 22}{27, 25, 56}{90, 88, 33}{33, 37, 47}


{94, 77, 38}{15, 81, 97}{7, 82, 28}{82, 86, 95}{94, 64, 76}{41, 12, 5}{80, 57, 15}{83, 75, 97}{50, 44, 6}{5, 44, 75}{64, 83, 12}{43, 78, 11}{55, 48, 76}{87, 62, 99}{35, 84, 90}{71, 34, 60}{85, 86, 63}{13, 79, 6}{23, 87, 32}{43, 37, 86}{29, 66, 35}{82, 47, 29}{33, 75, 7}{68, 98, 51}{34, 84, 9}{84, 43, 6}{32, 81, 79}{79, 27, 82}{43, 49, 13}{21, 60, 28}{11, 81, 69}{53, 26, 95}

{93, 34, 15}{93, 14, 12}{85, 6, 68}{62, 58, 33}{61, 15, 4}{28, 16, 66}{23, 42, 13}{93, 26, 44}{40, 96, 5}{34, 30, 29}{98, 12, 13}{33, 80, 31}{0, 91, 19}{72, 50, 23}{8, 2, 64}{40, 52, 67}{32, 53, 74}{56, 43, 18}{31, 61, 46}{87, 26, 28}{50, 4, 76}{57, 73, 26}{76, 82, 12}{92, 97, 42}{80, 86, 83}{26, 7, 49}{80, 88, 32}{76, 95, 67}{56, 98, 34}{2, 14, 23}{52, 54, 21}{51, 5, 15}

{25, 97, 50}{72, 19, 25}{28, 42, 38}{95, 70, 51}{48, 1, 31}{74, 66, 38}{69, 43, 53}{3, 77, 22}{24, 41, 53}{6, 71, 33}{44, 20, 42}{28, 46, 55}{38, 16, 95}{60, 24, 63}{73, 56, 55}{59, 22, 33}{20, 74, 73}{30, 76, 49}{23, 97, 78}{3, 26, 85}{38, 56, 9}{74, 6, 39}{17, 43, 29}{23, 25, 6}{69, 45, 37}{85, 88, 77}{38, 22, 30}{97, 67, 99}{13, 67, 1}{49, 40, 6}{24, 98, 99}{21, 2, 78}

{67, 23, 5}{10, 58, 22}{52, 39, 43}{57, 27, 59}{64, 9, 66}{88, 83, 61}{5, 80, 93}{79, 5, 95}{10, 4, 66}{9, 41, 28}{72, 8, 95}{15, 40, 19}{94, 55, 87}{13, 54, 84}{47, 16, 23}{54, 57, 16}{91, 99, 68}{85, 66, 19}{30, 58, 66}{45, 41, 99}{47, 17, 92}{85, 22, 40}{60, 12, 1}{28, 51, 81}{94, 33, 43}{69, 36, 63}{79, 66, 83}{19, 48, 43}{14, 78, 25}{72, 3, 91}

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Index

all-pair testing, 7

balanced incomplete block design, 13, 84

BIBD, see balanced incomplete block de-

sign

CA, see covering array

CARL, see covering array with row limit

CARLN , 26

ceiling function, 12

Chebyshev’s inequality, 22

combinatorial testing, 7

covering, 18, 81

cover number, 18

covering array, 20

cover number, 21

optimal, 21

covering array with row limit, 24

applications, 7

asymptotic size, 43, 49

cover number, 26

degree, 25

Johnson bound, 125

lower bound, 36

optimal, 26, 112, 113

order, 25

row limit, 25

size, 25

strength, 25

upper bound, 39, 41, 45

DDA, see deterministic density algorithm

deterministic density algorithm, 57

DGDD, see double group divisible design

double group divisible design, 17, 84, 90, 93,

103, 136

construction, 18

edge cover, 22

excess graph, see group divisible covering

design

floor function, 12

GDCD, see group divisible covering design

GDD, see group divisible design

GDPD, see group divisible packing design

graph covering problem, 29

group divisible covering design, 26

197

INDEX 198

cover number, 28, 112, 113

excess graph, 29

optimal, 28, 32, 70

group divisible design, 6, 14, 74, 76–78, 81,

84, 104, 107, 133, 137

group divisible packing design, 32

leave graph, 33

packing number, 32, 139

HGDD, see holey group divisible design

holey group divisible design, 16, 83, 89, 134,

135

hypergraph, 22, 49

IGDD, see incomplete group divisible de-

sign

incomplete group divisible design, 16, 81, 83

Johnson lower bound, 125

Johnson upper bound, 122, 126

leave graph, see group divisible packing de-

sign

Markov’s inequality, 22, 42

OA, see orthogonal array

order of a set, 12

orthogonal array, 15, 21

PA, see packing array

packing, 20

packing array, 31

packing array with row limit, 31

optimal, 31, 139

row limit, 31

packing arrays with row limit

asymptotic size, 120

upper bound, 119, 122, 124

packing number, 32, 139

pairwise balanced design, 13, 104

mandatory block, 13, 84, 104

parallel class, 13

PARL, see packing array with row limit

PBD, see pairwise balanced design

product construction, 57

resolvable design, 13

Rodl’s nibble, 22

Schonheim lower bound, 36, 68

Schonheim upper bound, 119, 126

second moment method, 22

TD, see transversal design

t-design, 13

transversal design, 15

UB1, 41

Wilson’s construction, 61, 71, 103, 134–136

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