c. camarero, c. mart nez and r. beivide -...

Introduction and Motivation Introduction to L-graphs Identifying codes Conclusions and Future Work

Identifying Codes over L-graphs

C. Camarero, C. Martınez and R. Beivide

University of Cantabria

Castell de Cardona 15 September 2011

C.Camarero et al. (U. of Cantabria) Identifying Codes over L-graphs Cardona 15 Sep 2011 1 / 23

• Thank you very much for the presentation.• I am going to present the work entitled Identifying codes over L-graphs• The outline of the talk is as follows ...

http://www.alumnos.unican.es/ccc66

http://www.unican.es/


Outline of the Talk

1 Introduction and Motivation

2 Introduction to L-graphs

3 Identifying codes over Z2 and L-graphs

4 Conclusions and Future Work


• We’ll begin introducing the identifying codes and giving the originalmotivation.

• Then we will introduce L-graphs, which are the four-degree Cayleygraphs over Abelian groups.

• Later we will show the identifying codes we have constructed over boththe integer plane and the L-graphs

• And we will end giving some Conclusions and possible lines of futurework.


Identifying codes in multiprocessor systems

Multiprocessor systems are modeled by a graph of nodes with theedges representing the communications.

Multiprocessor systems canhave faulty nodes. Hence weneed fault diagnosis.

Fault diagnosis can be approached by identifying codes.


• Multiprocessor systems are modeled by the graph with vertices the nodesand with the edges representing the communications between processors.

• For example, I show in the picture the topology of the Illiac IVsupercomputer, which is a twisted torus.

• The processors of a system can have faults and there is the problem oflocating them.

• This is the fault diagnosis problem and it can be approached usingidentifying codes.


Definition of Identifying codes

Let G = (V,E) be a graph. G induces a metric d, counting thenumber of edges in shortest paths.

The ball of radius r centered in v ∈ V isBr(v) = {w ∈ V |d(v, w) ≤ r}. We say that v r-covers theelements in its ball.

Given a code C ⊂ V , we define Kr(v) = C ∩Br(v).When Kr(v) 6= Kr(w) we say that v and w are r-separated.

Definition ([KarChaLev TIT98])

C is called r-identifying when the sets Kr(v) are nonempty anddifferent.


• Now we are going to give some definitions.• Given a graph we have the metric of the lengths of the paths.• Hence we can consider the ball centered in a vertex, which is the vertices

which are closer than a radius. And we say that the center of the ballr-covers the elements in the ball.

• A code is a subset of the vertices. We define K (key) as the codewords ina ball. When K is different for two words then we say that they arer-separated.

• Karpovski et al defined an r-identifying code as a code which has each Knonempty and different. This is, each word is univocally identified by asubset of the code.


Identification on grids

Identifying codes have been considered over several infinite grids:

We construct a wide family of identifying codes over the square grid.Previous work on the square grid: [HonLob JCT02],[Charon et al. EJC01] and [Charon et al EJC02].


• Identifying codes have been considered over several infinite grids:• As the square grid, the triangular, the king and the hexagonal.• (NEXT) Our construction is done over the square grid,• which many authors have already considered.


L-graphs

In applications like multiprocessor systems finite graphs arerequired.

We consider the graph obtained by Z2 modulo a lattice and call itan L-graph.

⇒modulo a

lattice


• But multiprocessor systems are finite, so we cannot keep using an infinitegrid.

• We can reduce the integer plane modulo a lattice, obtaining a new graphwhich we call L-graph.


Outline of the Talk


2 Introduction to L-graphsCayley graphsMultidimensional Circulant: L-graphsExamples




• Now I will give a more formal introduction to L-graphs.• We will begin giving Fiol’s definition of multidimensional circulant

graph, for which L-graphs are the bidimensional case.• We end the introduction with some examples of L-graphs.


Cayley graphs

Definition of Cayley Graphs

Definition

Given a group Γ and an adjacency set A ⊂ Γ, the Cayley graphG(Γ;A) is defined as the graph with vertices V = Γ and edges

E = {(v, v + g)|g ∈ A}

We assume:

1 A = −A (undirected)

2 0 6∈ A (simple)

3 〈A〉 = Γ (connected)

The infinite square grid can be defined as a Cayley Graph over Γ = Z2

and A =

{±(

10

),±(

01

)}.


• We begin as said giving the definition of Cayley Graphs.• The Cayley Graph over the group Γ with a adjacency set A is the graph

which has as vertices the elements of the group and as edges Γ operatedwith A.

• We will assume that the graph is undirected, simple and connected.Which translates in properties of the adjacency set.

• We note that the infinite square grid can be defined as a Cayley graphover Z2 (zeta squared).


Multidimensional Circulant: L-graphs

Congruences in Zn

Let M ∈Mn×n(Z) an n× n integer matrix.

Two vectors a, b ∈ Zn are congruent modulo M :

a ≡ b (mod M)⇔ a− b ∈MZn,

whereMZn = {c1m1 + . . .+ cnmn : ci ∈ Z}.

We denote as Zn/MZn, the group of integer vectors module M , whichhas |det(M)| elements when M is nonsingular.


• Now we give a special group, the one of the congruences over integervectors.

• Let M be an n-sided integer square matrix.• We say that two vectors are congruent when its difference is equal to M

times a vector.• We can consider now the quotient group, which has cardinal equal to the

determinant of the matrix.


Multidimensional Circulant: L-graphs

Definition of L-graphs

Definition ([Fiol DM95])

We call multidimensional circulant to the Cayley graph with groupV = Γ = Zn/MZn and adjacencies A ⊆ Zn.

For n = 2 we call them L-graphs.

We use adjacencies A = {±e1,±e2} =

{±(

10

),±(

01

)}. But this is

an assumption which doesn’t lose generality.

They will be denoted as L(M).


• Now we can pose the definition of the L-graphs.• Fiol defined multidimensional circulant graph as the Cayley graph over

the group of integer vector congruences.• They are actually equivalent to any Cayley graph over an Abelian group.• We are interested in the bidimensional case, to which we refer as

L-graphs.• We will use as adjacencies the orthonormal base, which does not suppose

any loss of generality.• Then we can simply denote them by L(M).


Examples

Example: Products of cycles

M =

(10 00 6

)Cartesian

A = {(±1, 0)t, (0,±1)t}

Kronecker

A = {±(1, 1)t,±(1,−1)t}


• Now we show as example of L-graphs two products of cycles.• We are going to multiply a cycle of length ten with another cycle of

length six.• For both products the generator matrix is the given diagonal matrix.• (NEXT) For the Cartesian product we take the orthonormal basis and

we obtain the torus graph.• (NEXT) For the Kronecker product we use another set of adjacencies

and obtain another L-graph.


Outline of the Talk



3 Identifying codes over Z2 and L-graphsOur construction over Z2

Identifying over L-graphs



• In this section we are going to see our construction.• We will begin constructing a family of identifying codes over the plane• and then we will see the conditions for them to be identifying also over

L-graphs.


Our construction over Z2

R-fixed balls

Definition

Let C be a code of Z2. The ball Br(v) is r-fixed by C if there exists atleast a codeword in each one of the next sets:

UR−r (v)∪BL+r (v), UL−r (v)∪BR+

r (v), BR−r (v)∪UL+r (v), BL−r (v)∪UR+

r (v)

UR−r

BL+r

BL−r

UR+r

UL−r

BR+rBR−r

UL+r


• We begin giving a new definition, the r-fixed balls.• First, we divide the boundary of the ball in several sets, two sets for each

quadrant.• We denote the quadrants using U for upper regions of the ball, B for the

bottom, L for left and R for right.• For each quadrant we have some nodes which are inside the ball touching

the boundary and denote them as minus. And the nodes outside the ball[next to them/touching it] are denoted by plus.

• We say that a ball is r-fixed if in each of the four unions of opposing setsthere are at least one codeword.

• Intuitively, being r-fixed can make the center r-separated from manyother words since a displacement would remove a codeword from a minusset or would add one from a plus set.



Lemma

For all v ∈ Z2, Br(v) is r-fixed by C = 〈(

tt+d

),(−(t+d)

t

)〉, where

r = t2 + dt+ d2−12 and gcd(2t, d) = 1.

Remark

In general a code with r-fixed balls is not r-identifying.


• Now we present the following result:• The balls centered in any vertex of the code generated by these vectors

(SENALAR?) are r-fixed by this radius when d is odd and t and d arecoprimes.

• We note that we are taking half of the lattices which are invariantagainst rotations.

• We illustrate the fact that the balls are r-fixed in the picture.• We see that the green codeword has its ball fixed by the four blue

codewords, each one in a different minus set.• (NEXT) Now we see another ball, made fixed by a different set of

codewords.• (NEXT) And other one, although there are only three blue codewords,

the upper one is in two of the given sets.• (NEXT) We also remark that in general that a code has all its ballsr-fixed does not imply necessarily that the code is r-identifying, andexamples can be easily found.



Sketch of the proof

Let v be a vertex. We prove that Br(v) is r-fixed, we look for acodeword c = v +

(xy

)∈ UR−r (v) ∪ BL+

r (v).1 When gcd(2t, d) = 1 we find x0, y0 with x0 + y0 = r.2 Other solutions x′ = x0 + λN , y′ = y0 + λN withN = 2r + 1 = t2 + (t+ d)2.

3 By Euclidean division we take λ such that 0 ≤ x′ < N .If 0 ≤ x′ ≤ r then c = v +

(x′

y′)

is in UR−r (v).

If r < x′ ≤ 2r so −r ≤ y′ < 0. Then c = v +(x′−Ny′)

is a codeword

in BL+r (v).

N


• We show here a sketch of the proof of the lemma.• For any vertex v (vi) we want to proof that its r-ball is r-fixed.• We find a codeword in one of the four sets, the other three are done by

considering the proper rotations.• From the coprimality condition we can find a codeword in the line which

goes through the upper right minus set (SENALAR?).• In this code we have for any codeword that the word reached by addingN in any dimension is also a codeword. Hence we have periodicallycodewords in the given line.

• Dividing then by N we get two cases, if x is lower than the radius wehave a codeword in the upper right minus set and otherwise we can finda codeword in the bottom left by subtracting N to the horizontalcomponent (SENALAR?).



Main result of the paper

Theorem

Let t, d be positive integers such that gcd(2t, d) = 1, then

C = 〈(

tt+ d

),

(−(t+ d)

t

)〉

is an r-identifying code over Z2 for r = t2 + dt+ d2−12 .


• We have said that having r-fixed balls was not sufficient to be anr-identifying code.

• But in the case of the code defined in the previous lemma this is actuallytrue.

• Thus we have given a wide family of identifying codes over the squareinfinite grid.



Sketch of the proof

We want to see that v ∈ V is r-separated from w = v +(xy

).

1 We can asume x > y ≥ 0.2 Since balls are fixed, we havec1 ∈ UL−r (v) ∪ BR+

r (v)a If c1 ∈ UL−r (v) Xb If c1 = v +

(a

a−r−1)∈ Kr(w) X

c If a > d−12 , we construct a codeword

in Kr(v) Xd Else, 1 ≤ a ≤ d−1

2 must be hold.

3 For BL−r (v) ∪ UR+r (v) we obtain

1 ≤ b ≤ t.4 Note that 2d and 3 cannot be hold

simultaneously.

v

w

2a

2b

2c

2d

3


• And now we give an idea of the proof of the theorem.• Given any pair of vertices v, w (vi, double vi) we want to see that they

are r-separated.• We can assume that their difference is in the first quadrant by doing the

proper rotation.• From the previous lemma we have that the ball in v is r-fixed, so we

have a codeword in the upper left minus set or the bottom right plus set.• If we had the codeword in the upper left we have them separated.• Otherwise it is in the bottom right set and we can parametrize it by a. If

it is K(w) we have them again separated.• Now we see than when a is greater than a constant we can find another

codeword which is in K(v).• But sometimes neither of them happen, like in the picture, and we have

bounded the possible values of the parameter a.• Now we can repeat the same procedure with the upper right plus and its

opposite, and we would bound the parameter b of other codeword aslower than t.

• But both conditions cannot be hold simultaneously, since there is no zeroin the region they define.



Over L-graphs

An identifying code over Z2 which is periodic will be identifying forsome L-graph. We need congruency (keeping density) and an lowerbound in its size.

Corollary

Let T =(

t −(t+d)t+d t

)and M such that

∃Q ∈M2×2(Z), M = TQ and L(M) hasa systole of length > 2r + 1. Then,

C = 〈(

tt+ d

),

(−(t+ d)

t

)〉

is an r-identifying code over L(M) for

r = t2 + dt+ d2−12 .


• We have already constructed a family of identifying codes over Z2, butwe said we were interested also in identifying over L-graphs. Now we willsay the conditions for these codes to be also identifying over L-graphs.

• The most obvious condition is that the code must be periodic.• Then, in the case of our lattice codes we need the generator matrix M to

be multiple of the matrix associated to the lattice. With this we get thatthe code is locally the same than in the plane, thus keeping its density.

• For the other condition we first look at the figure, where we depict anL-graph over the torus surface. The colored links are a minimalnontrivial cycle, which we call the L-graph systole.

• The second condition is that this systole must be greater than two timesthe radius. This means that a ball cannot overlap with itself.



Perfect Codes

Corollary

Let T =

(t −(t+ 1)

t+ 1 t

)and M such that exists Q ∈M2×2(Z)

satisfying M = TQ and that L(M) has a systole of lengthmin{d(0,Mγ) | 0 6= γ ∈ Z2} > 2r + 1. Then,

C = 〈(

tt+ 1

),

(−(t+ 1)

t

)〉

is both a t(t+ 1)-identifying code and t-perfect error correctingcode over L(M).


• An important special case of our family of identifying codes are theperfect error correcting codes.

• When we take d equal to one, we have the coprimality condition andhence that any perfect code over the integer plane or over an L-graphwith sufficiently large systole is both a t times t plus one identifying codeand a t perfect error correcting code.



Density of the construction

The density of our construction is:

D = limn→∞

|C ∩Qn||Qn|

=|C||V |

=

det(M)det(T )

det(M)=

1

det(T )=

1

2r + 1.

Which is worse than the current best construction of 25r

[HonLob JCT02].

However, our codes have a small covering radius and include all theperfect codes over Z2 and L-graphs. This is useful at the first stageof some applications.


• Now we analyze the density of the code we have constructed, which isproperty usually taken into account.

• The density is defined by the limit (SENALAR), which in the finite caseit simplifies as the number of codewords between the total of words,which in our case is simply the inverse of the determinant associated tothe lattice, this is, two times the radius.

• We need to note that this family has worse density than the best currentconstruction of 2/(5r) (two over five r) [for even radius].

• However, our codes have a small covering radius and include the perfectcodes over Z2 and L-graphs, which is useful at the first stage of someapplications, for example to determine if there is any fault in theproblem of diagnosis in multiprocessor systems.


Outline of the Talk






• And finally we show some conclusions


Conclusions and Future Work

We have presented lattice-based codes:

Wide family of identifying codes.Small covering, our family includes the perfect codes.Novel strategy: r-fixed balls.

Could this codes work in other meshes?:

triangular ,hexagonal ,king .


• We have presented lattice-based codes.• They are a wide family of identifying codes.• They have a small covering, and include all the perfect codes, which is

useful for applications.• We have used a novel strategy, the use of r-fixed balls, as intermediate

step to proof the identification.• As future work we can wonder if the same construction could work over

other grids, like the triangular, the hexagonal or the king.


Thanks!

Questions?


• Many thanks for your attention.

L-grafos Codigos perfectos Identification

Bibliography I

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I. Charon, I. S. Honkala, O. Hudry, Antoine Lobstein.General Bounds for Identifying Codes in Some Infinite Regular Graphs.Electr. J. Comb. 8(1): (2001).

G. D. Cohen, I. S. Honkala, A. Lobstein, G. Zemor.New Bounds for Codes Identifying Vertices in Graphs.Electr. J. Comb. 6: (1999)

M. A. Fiol.Congruences in Zn, Finite Abelian Groups and the Chinese Remainder Theorem.Discrete Mathematics 67(1): 101-105 (1987).

M. A. Fiol, J. L. Yebra, I. Alegre and M. Valero.Discrete Optimization Problem in Local Networks and Data Alignment.IEEE Trans. Comput. 36, 6 (Jun. 1987), 702-713.



Bibliography II

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A. Lobstein.Watching systems, Identifying, Locating-dominating and Discriminating Codes inGraphs.Available at http://www.infres.enst.fr/~lobstein/debutBIBidetlocdom.pdf

C. Martınez, R. Beivide and E. Gabidulin.Perfect Codes for Metrics Induced by Circulant Graphs.IEEE Transactions on Information Theory, Vol. 53, No. 9, pp. 3042-3052. September2007.

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Y. Ben-Haim, S. Gravier, A. Lobstein and J. Moncel.Adaptative Identification in Graphs.Journal of Combinatorial Theory, Series A, Volume 115, Issue 7, Pages 1114-1126,October 2008.


http://www.infres.enst.fr/~lobstein/debutBIBidetlocdom.pdf


Bibliography III

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Cristobal Camarero, Carmen Martınez and Ramon BeivideIdentifying Codes over L-graphsAccepted for presentation at the 3th International Castle Meeting on Coding Theoryand Applications (3ICMTA).2011. Barcelona, Spain.

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Bibliography IV

C. Camarero, C. Martınez, and R. Beivide.Symmetric L-networks.In 2010 International Workshop on Optimal Network Topologies, 2010.

M. A. Fiol.On Congruence in Zn and the Dimension of a Multidimensional Circulant.Discrete Mathematics 141 (1995) 123-134.

Michael Kaib and Claus P. Schnorr.The generalized Gauss reduction algorithm.J. Algorithms, 21(3):565–578, 1996.

C. Martınez, R. Beivide, E. Stafford, M. Moreto, E. M. Gabidulin.Modeling Toroidal Networks with the Gaussian Integers.IEEE Transactions on Computers. Vol. 57, No.8, 1046-1056. August 2008.

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Bibliography V

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Bibliography VI

Camara, J., Moreto, M., Vallejo, E., Beivide, R., Miguel-Alonso, J., Martinez, C.,Navaridas, J.Twisted torus topologies for enhanced interconnection networks.IEEE Transactions on Parallel and Distributed Systems 99(PrePrints) (2010).



Apendice

5 Introduccion a L-grafos

6 Codigos perfectos

7 Identification



Enlaces Ortonormales

det(A)×G(M ;A) ' G(adj(A)M ; det(A)I) ' m×G(M ′)

C12(3, 4) ' G(

(12 00 1

);

(3 41 1

)) ' G(

(4 00 3

))

01

2

3

4

56

7

8

9

10

11

0 3 6 9

4 7 10 1

8 11 2 5



Producto de Ciclos con enlaces Ortonormales

El producto de Kronecker de dos ciclos admite una expresion directacomo L-grafo con enlaces ortonormales:

Theorem ([MarCamBei ISIT10])

Ca × Cb es isomorfo a:

a ≥ b impares ⇒ L(M) donde M =

(a+b

2a−b

2a−b

2a+b

2

).

a ≥ b pares ⇒ Dos copias disjuntas de L(M), donde

M =

(a2−b2

a2

b2

).

a par, b impar ⇒ L(M), donde M =

(a2 −ba2 b

).

Para el resto de L-grafos tenemos un algoritmo que deja losenlaces ortonormales.



L-forma

El L-grafo asociado a M =

(a bc d

)puede verse como:

d

|b|

c

a

b ≤ 0 ≤ a, c, d→ suma de

rectangulos. L(

(6 −54 7

)).

d b

c

a

0 ≤ a, b, c, d→ diferencia de

rectangulos. L(

(10 64 8

)).



Codigos Perfectos

We denote the ball of radius t centered in a vertex v of L(M ;E) asBM (v, t) = {w ∈ Z2/MZ2 | dM (v, w) ≤ t}. Clearly, BM (v, t) has2t2 + 2t+ 1 elements.

Definition

A perfect t-error correcting code over an L-graph L(M ;A) is a subset Cof the vertex set such that the set of vertices is a disjoint union of ballsas follows:

Z2/MZ2 =⋃

c∈CBM (c, t).

The previous definition is equivalent to:

Lemma

C ⊂ Z2/MZ2 is a perfect t-error correcting code over L(M ;A) if and

only if it is a subset of | det(M)|2t2+2t+1

vertices with the propertydM (c, c′) ≥ 2t+ 1, for all c, c′ ∈ C such that c 6= c′.



Codigos sobre Gaussianos

El retıculo de los enteros Gaussianos ha sido usado para modelarconstelaciones tipo QAM en [?].

Codigo en el grafo Gaussianosobre Z[i]3+4i.

La distancia de los grafos Gaussianos nos da una metrica ycodigos correctores de errores sobre ellas [?].



Theorem

Let M ∈M2×2(Z) be a non singular matrix such that there existQ ∈M2×2(Z) with M = TQ (respectively M = T tQ). Then, T +MZ2

is a perfect t-error correcting code over the graph L(M ;E) (respectivelyT t + Z2/MZ2 ).

Proof. Let us prove the case in which M = TQ. Since the equality ofcardinalities is obtained by Corollary ??, next we prove that given anypair of different words of the code there is a distance between themgreater than 2t+ 1. In this aim, let Tγ, Tγ′ be such that Tγ 6= Tγ′

(mod M). Since dM (Tγ, Tγ′) = dM (T (γ − γ′), 0), it is enough to showthat dM (Tγ, 0) ≥ 2t+ 1 for any γ ∈ Z2 such that Tγ 6= 0 (mod M).



If dM (Tγ, 0) < 2t+ 1 we have that Tγ ≡(xy

)(mod M) such that

|x|+ |y| < 2t+ 1 is minimum. Hence,

Tγ −(xy

)= Mγ′ = TQγ′,

which implies Tγ − TQγ′ =(xy

), that is,(

xy

)= T (I2γ −Qγ′) = Tγ′′, with |x|+ |y| < 2t+ 1, which is not

possible by Lemma ??.



Remark

Let M such that M 6= TQ for any Q ∈M2×2(Z). Therefore, it isstraightforward that T + Z2/MZ2 = Z2/MZ2, which implies that it isnot a perfect t-error correcting code. Clearly, any perfect codecontaining another vertex c different to zero is a traslation of the onecontaining vertex zero: c+ T (mod M).

As a direct consequence of the previous results and remark we havethat:

Corollary

L(M ;E) has a perfect t-error correcting code if and only if there existsQ ∈M2×2(Z) such that M = TQ (or M = T tQ).



Idea de la identificacion

Nos interesa la mınima densidad =⇒ Montones de artıculos sobrecotas de la densidad mınima.



Let t, d ≥ 0 be two integers such that gcd(2t, d) = 1 and let us define

r = t2 + dt+ d2−12 . We consider the code C = 〈

(t

t+d

),(−(t+d)

t

)〉 of the

infinite mesh. We denote also by C =

(t −(t+ d)

t+ d t

)the matrix

associated to the code. If N is the cardinal of Z2/CZ2, we have thatN = t2 + (t+ d)2 = 2r + 1. Since(

t −(t+ d)t+ d t

)(t+ dt

)=

(0N

)and (

t −(t+ d)t+ d t

)(t

−t− d

)=

(N0

)we obtain that if

(xy

)∈ C then ∀m,n ∈ Z,

(x+mNy+nN

)∈ C. We will use

this fact to prove the following:



Lemma

For all v ∈ V , Br(v) is r-fixed by C〈(

tt+d

),(−(t+d)

t

)〉, where

r = t2 + dt+ d2−12 and gcd(2t, d) = 1.

Proof. We will proof that for each v ∈ V there exists a codeword c inthe set UR−r (v) ∪ BL+

r (v). The other three codewords can be obtainedby making the suitable rotations. Note that, if c ∈ UR−r (v) ∪ BL+

r (v)

and R =

(0 −11 0

)then Rc is codeword in UL−r (Rv) ∪ BR+

r (Rv), R2c

is codeword in BL−r (R2v) ∪ UR+r (R2v) and R3c is codeword in

BR−r (R3v) ∪ UL+r (R3v).

Since c must be in C, there must exist integers a, b such thatc =

(at−b(t+d)a(t+d)+bt

). Now, if v =

(v1v2

), let us denote

(xy

)= c− v. Then, we

obtain the following system of Diophantine equations:{x = at− b(t+ d)− v1

y = a(t+ d) + bt− v2



Therefore, x+ y = a(2t+ d)− bd− v1 − v2. By hypothesis,gcd(2t+ d, d) = gcd(2t, d) = 1, which implies that there exist a0, b0 ∈ Zsuch that a0(2t+ d)− b0d = r+ v1 + v2, thus there also exist x0, y0 ∈ Zsuch that x0 + y0 = r. The set of solutions is{a′ = a0 + dλ, b′ = b0 + (2t+ d)λ | λ ∈ Z. Hence,x′ = a0t−b0(t+d)+tdλ−(t+d)(2t+d)λ = a0t−b0(t+d)−Nλ = x0+λNand in the same way y′ = y0 + λN .Now, we set λ such that 0 ≤ x′ < N = 2r + 1, so 0 ≤ x′ ≤ 2r. Now,

If 0 ≤ x′ ≤ r then we have that 0 ≤ y′ ≤ r. Therefore, taking(xy

)=(x′y′)

we obtain that c = v +(xy

)is in UR−r (v).

Otherwise, we have that r < x′ ≤ 2r so −r ≤ y′ < 0. Therefore,taking

(xy

)=(x′−Ny′)

we have that x+ y = r −N = −r − 1 and−r − 1 < x ≤ −1, that is −r ≤ x < 0, thus obtaining thatc = v +

(xy

)is a codeword in BL+

r (v).



Theorem

Let t, d be positive integers such that gcd(2t, d) = 1 and

r = t2 + dt+ d2−12 . Then, C = 〈

(t

t+d

),(−(t+d)

t

)〉 is a r-identifying code

in Z2.

Proof. By previous lemma we have that all r-balls are r-fixed.However, we know that this is not enough for being a r-identifyingcode. Therefore, we will next prove that any v ∈ V is r-separated fromw = v +

(xy

), for any

(xy

)∈ Z2. Note that, if |x| = |y| this is

straightforward since the balls are r-fixed. Then, by symmetry, we canassume without loss of generality that x > y ≥ 0.If there is a codeword in UL−r (v) then it is straightforward that theyare r-separated. Otherwise, we have a codeword c1 ∈ C ∩ BR+

r (v)



which is not in Kr(v). If c1 is in Kr(w) we have finished. Asc1 ∈ BR+

r (v) it can be expressed as c1 = v +(

aa−r−1

). Then

c1 +(−(t+d)

t

)= v +

(a−(t+d)a+t−r−1

)is a codeword which is further from Kr(w) than c1. And whena > d−1

2 , it is in Kr(v).If we do the same than in previous case, but for BL−r (v) ∪ UR+

r (v), weobtain c2 = v +

(b

r−b+1

). Then, we consider

c2 +(−(t+d)

t

)−(

tt+d

)= c2 +

(−2t−d−d

)= v +

(b+t

r−b+t+d+1

)which is further from Kr(w) than c2 and when b > t it is in Kr(v).To finish we need to prove that the case when both codewords c1, c2

are in the left ranges never happen. For that, when 1 ≤ a ≤ d−12 and

1 ≤ b ≤ t, with c1, c2 ∈ C should be c = c1 − c2 ∈ C but we will obtain a



contradiction. We have that c = c1 − c2 =(

a−ba+b−1−N

)=(x′y′)

is in therectangle defined by:

{1− t ≤ x′ < d− 1

2−1 < d, 1−N ≤ y′ ≤ t+ d− 1

2−1−N < t+d−N}

which does not include the point(

0N

). This rectangle is a subset of

{1− 2t ≤ x′ ≤ −t,N ≤ y′ ≤ N + t− 1}∪{1− t ≤ x′ ≤ d,N ≤ y′ ≤ N + t+ d− 1}

which is a set of representatives of Z2/CZ2 (see [JorPot MM65]) andcontains

(0N

). As a consequence, all its elements are different.

Therefore, c is not congruent with(

00

)and thus it is not a codeword,

which is a contradiction.



Corollary

Let T =

(t −(t+ d)

t+ d t

)and M be such that there exists

Q ∈M2×2(Z) such that M = TQ and L(M) have a systolic lengthmin{d(0,Mγ) | 0 6= γ ∈ Z2} > 2r + 1. Then, C = 〈

(t

t+d

),(−(t+d)

t

)〉 is

an r-identifying code in L(M) for r = t2 + dt+ d2−12 .

Remark

Remember that the systolic length min{d(0,Mγ) | 0 6= γ ∈ Z2} > 2r+ 1is the length of the shortest non-trivial cycle in L(M).


c. camarero, c. mart nez and r. beivide -...

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