a short course at tamkang university taipei, taiwan, r.o.c. march 7-9 2006 an application of coding...

A Short Course at Tamkang University Taipei, Taiwan, R.O.C. March 7-9 2006

An Application of Coding Theory into Experimental Design

Shigeichi HirasawaDepartment of Industrial and

Management Systems Engineering,

School of Science and Engineering ,

Waseda University

No.2

１． Introduction序論

No.3

1.1 　Abstract

Orthogonal Arrays (OAs)

Error-Correcting Codes (ECCs)

Experimental Design Coding Theory

・ relations between OAs and ECCs

・ the table of OAs and Hamming codes

・ the table of OAs + allocation

table of OA L8 etc.

Hamming codes,

BCH codes

RS codes etc.

close relation

実験計画符号理論

直交配列

直交表 L ８

No.4

1.2 　 Outline

１． Introduction

２． Preliminary

３． Relation between ECCs and OAs

４． Conclusion

序論

準備

結論

No.5

準備2 ． Preliminary

No.6

実験計画法Experimental Design

No.7

２．１　 Experimental Design ( 実験計画法 )

・ Factor A (materials)A0 （ A company ）， A1 （ B

company ）

・ Factor B （ machines ）B0 （ new ）， B1 （ ol

d ）

・ Factor C （ temperatures ）C0 （ 100℃ ）， C1 （

200℃ ）

a Ratio of Defective Products

Ex.)

・ How the level of factors affects a ration of defective products ?

・ Which is the best combination of levels ?

要因 A

要因 B

要因 C

2.1.1 　 Experimental Design

No.8

Complete Array

Ａ　　Ｂ　　Ｃ０　　０　　０

０　　０　　１

０　　１　　０

０　　１　　１

１　　０　　０

１　　０　　１

１　　１　　０

１　　１　　１

Experiment ①

　　　②

　　　③

　　　④

　　　⑤

　　　⑥

　　　⑦

　　　⑧

experiment with A0,B0,C0

experiments with all combination of levels

完全配列

実験

No.9

Ａ　　Ｂ　　Ｃ０　　０　　０

０　　１　　１

１　　０　　１

１　　１　　０

　　　②

　　　③

　　　④

Orthogonal Array (OA) : OA(M, n, s,τ) (s=2)

100 110

000 010

011

111101

001

strength 　 τ=2

subset (subspace) of complete array

Experiment ①

every 2 columns contains each 2-tuple exactly same times as row

直交配列

部分空間

強さ

No.10

Ａ　　Ｂ　　Ｃ０　　０　　０

０　　１　　１

１　　０　　１

１　　１　　０

　　　②

　　　③

　　　④ 100 110

000 010

011

111101

001Experiment ①



直交配列

部分空間

strength 　 τ=2


強さ

No.11

Ａ　　Ｂ　　Ｃ０　　０　　０

０　　１　　１

１　　０　　１

１　　１　　０

　　　②

　　　③

　　　④ 100 110

000 010

011

111101

001Experiment ①



直交配列

部分空間

strength 　 τ=2


強さ

No.12

Ａ　　Ｂ　　Ｃ０　　０　　０

０　　１　　１

１　　０　　１

１　　１　　０

　　　②

　　　③

　　　④ 100 110

000 010

011

111101

001Experiment ①



直交配列

部分空間

strength 　 τ=2


強さ

No.13

Parameters of OAs

・ the number of factors n・ the number of runs　 M・ strength 　τ=2t

A 　　B 　　 C①

　　　②　　　③　　　④

０　　０　　００　　１　　１１　　０　　１１　　１　　０

the number of factors n=3

the number of runs 　M=4

strength τ=2

Construction problem of OAs is to construct OAs with as few as possible number of runs, given the number of factors and strength (n,τ 　→　 min M)

this can treat t-th order interaction effect

trade off

2.1.2 　 Construction Problem of OAs

因子数

実験回数

強さ

因子数

実験回数

強さ

No.14

Generator Matrix of an OA : G

Ex.) orthogonal array ｛ 000 , 011 , 101 , 110 ｝

（○，○，○） = （□，□）

０　１　１１　０　１

OA each k-tuple (k=2) based on｛ 0,1 ｝ 2 2k=M

generator matrix G

A B CA B C

2.1.3 　 Generator Matrix 　 ( 生成行列 )

To construct OAs is to construct generator matrix

No.15

orthogonal array ｛ 000 , 011 , 101 , 110 ｝

（０，０，０） = （０，０）

０　１　１

A B CA B C

１　０　１

OA generator matrix G

Ex.)




each k-tuple (k=2) based on｛ 0,1 ｝ 2 2k=M

No.16


（０，１，１） = （１，０）

０　１　１

A B CA B C

１　０　１

OA


generator matrix G

Ex.)




No.17


（１，０，１） = （０，１）

０　１　１

A B CA B C

１　０　１


Ex.)





No.18


（１，１，０） = （１，１）

０　１　１

A B CA B C

１　０　１


Ex.)





No.19

Parameters of OAs and Generator Matrix : G

orthogonal arrays ｛ 000 , 011 , 101 , 110 ｝

・ the number of factors n=3・ the number of runs M=4・ strength τ=2

０　１　１１　０　１

G =

3

2


the number of runs M=22

any 2 columns are linearly independent

strength 　 τ=2

Ex.)

No.20

Parameters of OAs and Generator Matrix


０　１　１１　０　１

G =

3

2


Ex.)

0

1

1

0+

0

0≠




strength 　 τ=2

No.21



０　１　１１　０　１

G =

3

2


Ex.)

0

1

1

1+

0

0≠




strength 　 τ=2

No.22



０　１　１１　０　１

G =

3

2


Ex.)

1

0

1

1+

0

0≠




strength 　 τ=2

No.23

OAs and ECCs [HSS ‘99]

G =

n

m

OAs with generator matrix G ECCs with parity check matrix G

・ the number of factors n・ the number of runs M=2m

・ strength τ=2t

・ code length n・ the number of information symbols k=n-m

・ minimum distance d=2t + 1 this can correct all t errors

any τ=2t columns are linearly independent

this can treat all t-th order interaction effect

No.24

Coding Theory

No.25

２．２　 Coding Theory （符号理論）2.2.1 Coding Theory

techniques to achieve reliable communication over noisy channel (ex. 　 CD, cellar phones etc.)

0 → 000

1 → 111

0 000 100 0

Ex.)

encoder channel decoder

noise

codewords符号語

No.26

Error-Correcting Codes

subspace of linear vector space

100 110

000 010

011

111101

001

０

１

０００１１１

codeword

Ex.)

誤り訂正符号

部分空間

符号語

No.27

・ code length n

・ the number of information symbols 　k

・ minimum distance d=2t + 1 this can correct t errors

trade off

００００

１１１１

the number of information symbols 　 k=1

minimum distance d=3

this can correct 1 error

2.2.2 　 Construction Problem of ECCs : (n, k, d) codeParameters of ECCs

Construction problem of ECCs is to construct ECCs with as many as possible number of information symbols, given the code length and minimum distance （ n, d → max k ）

符号長

情報記号数

最小距離

No.28

Parity Check Matrix of ECCs

Ex.) 　　 (3,1,3) code ｛ 000 , 111 ｝

parity check matrix H =0 1 1 1 0 1

0 1 1 1 0 1

000

= 00

0 1 1 1 0 1

111

= 00

codeword

To construct of linear codes is to construct parity check matrix

2.2.3 　 Parity Check Matrix

HxT=0

No.29

Parameters of ECCs and Parity Check Matrix

・ code length n=3・ the number of information symbols k=1

・ minimum distance d=3

０　１　１１　０　１

H =

3

2

code length 　n=3

the number of information symbols k=3 - 2

any d-1=2 columns are linearly independent

minimum distance　 d=2 +1

Ex.) 　　 (3,1,3) code ｛ 000 , 111 ｝

No.30


G =

n

m

OAs with generator matrix G ECCs with parity check matrix G

・ the number of factors n・ the number of runs 　M=2m

・ strength τ=2t

・ code length n・ the number of information symbols n-m


any d-1=2t columns are linearly independent

this can treat all t order interaction effect

No.31

3 ． Relation Between OAs and ECCs

関係

No.32

3.1 　 OAs and ECCs

０　１　１１　０　１

G =

100 110

000 010

011

111101

001

100 110

000 010

011

111101

001

OA with generator matrix G ECC with parity check matrix G

No.33


G =

n

m

OAs with generator matrix G

・ the number of factors n・ the number of runs M=2m

・ strength τ=2t

・ code length n・ the number of information symbols k=n-m


any 2t columns are linearly independent

this can treat all t order interaction effect

ECCs with parity check matrix G

No.34

Table of OAs and Hamming Codes直交表

No.35

3.2 　 Matrix 　 in which any 2 columns are linearly

an OA with strength τ=2 ， a linear code with minimum distance

independent ①

０　０　０　１　１　１　１０　１　１　０　０　１　１１　０　１　０　１　０　１

001

111

・・・

G = 3

n=7

No.36


０　０　０　１　１　１　１０　１　１　０　０　１　１１　０　１　０　１　０　１

G = 3

independent ①

001

010

+ ≠000

an OA with strength τ=2 ， a linear code with minimum distance 0

01

111

・・・

n=7

No.37


an OA with strength 2 ， a linear code with minimum distance

０　０　０　１　１　１　１０　１　１　０　０　１　１１　０　１　０　１　０　１

G = 3

independent ②

・ table of OA L8

・（ 7,4,3 ） Hamming code

the number of factors 7 ， the number of runs 8 ， strength 2

code length 7, the number of information symbols 4, minimum distance 3

001

111

・・・

n=7

No.38


an OA with strength 2 ， a linear code with minimum distance

independent ①

・ table of OA L16

・（ 15,11,3 ） Hamming code

the number of factors 15 ， the number of runs 16 ， strength 2

code length 15, the number of information symbols 11, minimum distance 3

０　０　０　１　１　１　１　０　０　０　０　１　１　１　１０　１　１　０　０　１　１　０　０　１　１　０　０　１　１１　０　１　０　１　０　１　０　１　０　１　０　１　０　１

G =

０　０　０　０　０　０　０　１　１　１　１　１　１　１　１

4

No.39

Table of OAs + allocation直交表割付

No.40

1 　 2 　 3 　 4 　5 　 6 　 7 ①

　　　②　　　③　　　④　　　⑤　　　⑥　　　⑦　　　⑧

3.3 　 Example 　（ Allocation to L8 ）

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

L8 Linear Graph

１

２４

３５

６

７

線点図

No.41

1 　 2 　 3 　 4 　5 　 6 　 7

　　　②　　　③　　　④　　　⑤　　　⑥　　　⑦　　　⑧

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

１

２４

３５

６

７

factor A

BD

EA×B

BA D EC

C

3.3 　 Example 　（ Allocation to L8 ）

L8 Linear Graph

①

線点図

No.42

3.4 　 Construction Problem （ General Case ）

Special Case

・ the number of factors n=5

・ strength τ=4

an OA with as few as possible of runs

factors 　A,B,C,D,E

this can treat all L=2 order interaction effects （ A×B,A×C, ・・・ ,D×E ）

General Case

・ the number of factors n=5

・　？ this can treat partial 2order interaction effects （ A×B ）

Ex.)

an OA with as few as possible of runs

No.43

3.5 　 Generator Matrix （ General Case ）

Special Case （ A×B,A×C, ・・・ ,D×E ）

General Case （ A×B ）

A B C D Egenerator matrix G =


A B C D E

・ any 4 columns are linearly independent

・ any 3 columns which contain A, B are linearly independent

factors 　A,B,C,D,E

Ex.)

generator matrix G =

No.44

3.6 　 Meaning of allocation

Generator Matrix of L8 Projective Geometry （ Linear Graph ）

0 　 0 　 0 　 1 　1 　 1 　 10 　 1 　 1 　 0 　0 　 1 　 11 　 0 　 1 　 0 　1 　 0 　 1

001

010 100

011 101

110

111

No.45

0 　 0 　 0 　 1 　1 　 1 　 10 　 1 　 1 　 0 　0 　 1 　 11 　 0 　 1 　 0 　1 　 0 　 1

001

010 100

011 101

110

BA D ECfactor A

BD C

E

if C, D, E are not allocated to this column, any 3 columns which contain A, B are linearly independent

A×B

111

3.7 　 Meaning of allocation

Generator Matrix of L8 Projective Geometry （ Linear Graph ）

No.46

4 ． Conclusion

No.47

4.1 　Conclusion

１． Construction problems

ECCs : n, d → max k

OAs : n, τ → min M

２． A generator matrix of OAs is equal to a parity check matrix of ECCs.

３． Relations of each columns in construction problems of OAs are more complicate than in those of ECCs.

No.48

参考文献）[Taka79] I.Takahashi, “Combinatorial Theory and its Applications (in Japanese), ” Iwanami shoten, Tokyo, 1979

[HSS99] A.S.Hedayat ， N.J.A.Sloane ， and J.Stufken ，“　Orthogonal Arrays : Theory and Applications ，” Springer ， New York ， 1999 ．

[SMH05] T.Saito, T.Matsushima, and S.Hirasawa, “A Note on Construction of Orthogonal Arrays with Unequal Strength from Error-Correcting Codes,” to appear in IEICE Trans. Fundamentals.

[MW67] B.Masnic, and J.K.Wolf, “On Linear Unequal Error protection Codes,” IEEE Trans. Inform Theory, Vol.IT-3, No.4, pp.600-607, Oct.1967

a short course at tamkang university taipei, taiwan, r.o.c. march 7-9 2006 an application of coding...

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