The 2006 International Seminar of E-commerce Academic and Application Research Tainan, Taiwan, R.O.C. March 1-2,2006 An Application of Coding Theory into

The 2006 International Seminar of E-commerceAcademic and Application Research

Tainan, Taiwan, R.O.C. March 1-2,2006

An Application of Coding Theory into Experimental Design

Shigeichi HirasawaDepartment of Industrial and

Management Systems Engineering,

School of Science and Engineering ,

Waseda University

- Construction Methods for Unequal Orthogonal Arrays -

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． Unequal Error Protection Codes and

OAs

５． Examples of OAs with Unequal Strength

６． 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 ． Unequal Error Protection Codes and OAs

No.47

4.1 　 Unequal Error Protection CodesUnequal Error Protection Codes

(○ , ○ , ○) codeword

error protection levels are equal in each position of a codeword

(○ , ○ , ○) codeword

t+1 t t

error protection level t in each position

error protection level (t1,t2,t3) = (t+1, t, t)

t t t


Unequal Error Protection Codes

error protection levels are unequal in each position of a codeword

→ 　 minimum distance d=2t +1

→ 　 separation di=2ti +1

this is used to send numerical data

No.48

Construction Problem of Unequal Error Protection Codes



・ code length n code with as many as possible number information symbols M

・ code length n code with as many as possible number information symbols M

・ minimum distance

・ minimum distance d

(d1,d

2, ・・・ ,dn)

No.49

Unequal Error Protection Codes and Parity Check Matrix



H= 1 i n･･･

･･･minimum distance d

H= 1 i n

any di-1 columns that contain i-th column are linearly independent

separation 　 (d1 , 　･･･ , di , 　･･･ ,

dn )

･･･

･･･

any d-1 columns are linearly independent

No.50

4.2 　 Classification of OA

① 　 OA （ General Case ）

② 　 OA with unequal strength (τ1, τ2, ・・・ , τn)

→ 　 this can treat all τi/2 = 　 ti-th order interaction effect that contain i-th factor

③ 　 OA with (equal) strength τ

→ 　 this can treat all τ/2 = t-th order interaction effect

Ex.) 　 (factor 　 A,B,C)

① 　 A×B

② 　 A×B, A×C

③ 　 A×B, A×C, B×C

①

③

②

No.51

4.2 　 Classification of OA

① 　 OA （ General Case ）

② 　 OA with unequal strength (τ1, τ2, ・・・ , τn)

Unequal Protection Codes

③ 　 OA with (equal) strength τ


①

③

②Unequal Protection Codes


No.52

5. Examples of OAs with unequal strength

No.53

OAs from ECC and Unequal Error Protection Codes

OAs from BCH Codes

OAs from Unequal Error Protection Codes

・ number of factors 　 63

・ number of experiments 　 218

・ strength 6

・ number of factors 　 63

・ number of experiments 　 212

・ strength (6, 6, ・・・ , 6, 4, 4, ・・・ , 4)

→this can treat all 3-rd order interaction effect

→this can treat partial 3-rd order interaction effect

7

No.54

6 ． Conclusion

No.55

6.1 　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.56

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

Documents

The 2006 International Seminar of E-commerce Academic and Application Research Tainan, Taiwan, R.O.C. March 1-2,2006 An Application of Coding Theory into