An Application of Coding Theory into Experimental Design

An Application of Coding Theory into Experimental Design A Short Course at Tamkang University Taipei, Taiwan, R.O.C. March 7-9 2006 Shigeichi Hirasawa Department of Industrial and Management Systems Engineering, School of Science and Engineering , Waseda University

１．Introduction 序論

1.1　Abstract 符号理論実験計画 Experimental Design Coding Theory 直交配列 Error-Correcting Codes (ECCs) Orthogonal Arrays (OAs) close relation Hamming codes, BCH codes RS codes etc. 直交表 L８ table of OA L8 etc. ・ relations between OAs and ECCs ・ the table of OAs and Hamming codes ・ the table of OAs + allocation

1.2　Outline １．Introduction ２．Preliminary ３．Relation between ECCs and OAs ４．Conclusion 序論準備結論

2．Preliminary 準備

Experimental Design 実験計画法

２．１　Experimental Design (実験計画法) 2.1.1Experimental Design Ex.) 要因A ・ Factor A (materials) A0（A company），A1（B company） a Ratio of Defective Products 要因B ・Factor B （machines） B0（new），B1（old）要因C ・Factor C （temperatures） C0（100℃），C1（200℃）・How the level of factors affects a ration of defective products ? ・Which is the best combination of levels ?

完全配列 Complete Array experiments with all combination of levels Ａ　　Ｂ　　Ｃ experiment with A0,B0,C0 実験０　　０　　００　　０　　１０　　１　　００　　１　　１１　　０　　０１　　０　　１１　　１　　０１　　１　　１ Experiment ① 　　　② 　　　③ 　　　④ 　　　⑤ 　　　⑥ 　　　⑦ 　　　⑧

直交配列 Orthogonal Array (OA) : OA(M, n, s,τ) (s=2) 部分空間 subset (subspace) of complete array Ａ　　Ｂ　　Ｃ 011 001 Experiment ① ０　　０　　００　　１　　１１　　０　　１１　　１　　０ 101 111 　　　② 　　　③ 000 010 　　　④ 100 110 強さ strengthτ=2 every 2 columns contains each 2-tuple exactly same times as row

2.1.2Construction Problem of OAs 因子数 the number of factors n=3 Parameters of OAs A　　B　　C 因子数・the number of factors n ① ０　　０　　０実験回数　　　② ０　　１　　１・the number of runsM 実験回数 the number of runsM=4 強さ　　　③ １　　０　　１・strengthτ=2t trade off 　　　④ １　　１　　０ this can treat t-th order interaction effect 強さ 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)

2.1.3Generator Matrix(生成行列) Generator Matrix of an OA : G Ex.) orthogonal array ｛ 000 , 011 , 101 , 110 ｝ A B C A B C ０　１　１（○，○，○） = （□，□）１　０　１ OA each k-tuple (k=2) based on｛0,1｝22k=M generator matrix G To construct OAs is to construct generator matrix

2.1.3Generator Matrix(生成行列) Generator Matrix of an OA : G Ex.) orthogonal array ｛ 000 , 011 , 101 , 110 ｝ A B C A B C ０　１　１（０，０，０） = （０，０）１　０　１ OA each k-tuple (k=2) based on｛0,1｝22k=M generator matrix G To construct OAs is to construct generator matrix

2.1.3Generator Matrix(生成行列) Generator Matrix of an OA : G Ex.) orthogonal array ｛ 000 , 011 , 101 , 110 ｝ A B C A B C ０　１　１（０，１，１） = （１，０）１　０　１ OA each k-tuple (k=2) based on｛0,1｝22k=M generator matrix G To construct OAs is to construct generator matrix

2.1.3Generator Matrix(生成行列) Generator Matrix of an OA : G Ex.) orthogonal array ｛ 000 , 011 , 101 , 110 ｝ A B C A B C ０　１　１（１，０，１） = （０，１）１　０　１ OA each k-tuple (k=2) based on｛0,1｝22k=M generator matrix G To construct OAs is to construct generator matrix

2.1.3Generator Matrix(生成行列) Generator Matrix of an OA : G Ex.) orthogonal array ｛ 000 , 011 , 101 , 110 ｝ A B C A B C ０　１　１（１，１，０） = （１，１）１　０　１ OA each k-tuple (k=2) based on｛0,1｝22k=M generator matrix G To construct OAs is to construct generator matrix

Parameters of OAs and Generator Matrix : G Ex.) orthogonal arrays ｛ 000 , 011 , 101 , 110 ｝ the number of factors n=3 3 ・the number of factors n=3 ０　１　１ G = ・the number of runs M=4 2 １　０　１ the number of runs M=22 ・strength τ=2 any 2 columns are linearly independent strengthτ=2

Parameters of OAs and Generator Matrix Ex.) orthogonal arrays ｛ 000 , 011 , 101 , 110 ｝ the number of factors n=3 3 ・the number of factors n=3 ０　１　１ G = ・the number of runs M=4 2 １　０　１ the number of runs M=22 ・strength τ=2 any 2 columns are linearly independent 0 1 0 strengthτ=2 + ≠ 1 0 0

OAs and ECCs[HSS ‘99] n G = m any τ=2t columns are linearly independent OAs with generator matrix G ECCs with parity check matrix G ・the number of factors n ・code length n ・the number of runs M=2m ・the number of information symbols k=n-m ・strength τ=2t ・minimum distance d=2t + 1 this can treat all t-th order interaction effect this can correct all t errors

Coding Theory

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.) Ex.) 符号語 codewords 0 → 000 1 → 111 noise 000 100 0 0 encoder decoder channel

誤り訂正符号 Error-Correcting Codes 部分空間 subspace of linear vector space Ex.) 011 001 ００００ 101 111 １１１１ 000 010 符号語 100 110 codeword

2.2.2Construction Problem of ECCs : (n, k, d) code the number of information symbolsk=1 Parameters of ECCs 符号長・code length n ００００情報記号数・the number of information symbolsk １１１１ trade off 最小距離 minimum distance d=3 ・minimum distance d=2t + 1 this can correct 1 error this can correct t errors 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）

2.2.3Parity Check Matrix Parity Check Matrix of ECCs Ex.)(3,1,3) code ｛ 000 , 111 ｝ 0 1 1 1 0 1 parity check matrix H = codeword 000 111 0 1 1 1 0 1 0 1 1 1 0 1 00 00 = = HxT=0 To construct of linear codes is to construct parity check matrix

Parameters of ECCs and Parity Check Matrix code lengthn=3 Ex.)(3,1,3) code ｛ 000 , 111 ｝ the number of information symbols k=3 - 2 ・code length n=3 3 ０　１　１・the number of information symbols k=1 H = 2 １　０　１・minimum distance d=3 any d-1=2 columns are linearly independent minimum distanced=2 +1

OAs and ECCs [HSS ‘99] n G = m any d-1=2t columns are linearly independent OAs with generator matrix G ECCs with parity check matrix G ・the number of factors n ・code length n ・the number of runsM=2m ・the number of information symbols n-m ・strength τ=2t ・minimum distance d=2t + 1 this can treat all t order interaction effect this can correct all t errors

3．Relation Between OAs and ECCs 関係

3.1OAs and ECCs ０　１　１ G = １　０　１ ECC with parity check matrix G OA with generator matrix G 011 011 001 001 101 101 111 111 000 010 000 010 100 110 100 110

OAs and ECCs [HSS ‘99] n G = m any 2t columns are linearly independent OAs with generator matrix G ECCs with parity check matrix G ・the number of factors n ・code length n ・the number of runs M=2m ・the number of information symbols k=n-m ・strength τ=2t ・minimum distance d=2t + 1 this can treat all t order interaction effect this can correct all t errors

Table of OAs and Hamming Codes 直交表

3.2Matrixin which any 2 columns are linearly independent ① an OA with strength τ=2，a linear code with minimum distance 0 1 ・・・ 0 1 1 1 n=7 ０　０　０　１　１　１　１ G = ０　１　１　０　０　１　１ 3 １　０　１　０　１　０　１

3.2Matrixin which any 2 columns are linearly independent ① an OA with strength τ=2，a linear code with minimum distance 0 1 ・・・ 0 1 1 1 n=7 ０　０　０　１　１　１　１ G = ０　１　１　０　０　１　１ 3 １　０　１　０　１　０　１ 0 0 0 0 + 1 ≠ 0 1 0 0

3.2Matrixin which any 2 columns are linearly independent ② an OA with strength 2，a linear code with minimum distance 0 1 ・・・ 0 1 1 1 n=7 ０　０　０　１　１　１　１ G = ０　１　１　０　０　１　１ 3 １　０　１　０　１　０　１・table of OAL8 the number of factors 7，the number of runs 8，strength 2 ・（7,4,3）Hamming code code length 7, the number of information symbols 4, minimum distance 3

3.2Matrixin which any 2 columns are linearly independent ① an OA with strength 2，a linear code with minimum distance ０　０　０　０　０　０　０　１　１　１　１　１　１　１　１０　０　０　１　１　１　１　０　０　０　０　１　１　１　１ G = 4 ０　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・table of OAL16 the number of factors 15，the number of runs 16，strength 2 ・（15,11,3）Hamming code code length 15, the number of information symbols 11, minimum distance 3

Table of OAs + allocation 直交表割付

3.3Example　（Allocation to L8） 線点図 L8 Linear Graph 1　2　3　4　5　6　7 １ ① 0　0　0　0　0　0　0 　　　② 0　0　0　1　1　1　1 ７３５　　　③ 0　1　1　0　0　1　1 　　　④ 0　1　1　1　1　0　0 ６２４　　　⑤ 1　0　1　0　1　0　1 　　　⑥ 1　0　1　1　0　1　0 　　　⑦ 1　1　0　0　1　1　0 　　　⑧ 1　1　0　1　0　0　1

3.3Example　（Allocation to L8） 線点図 L8 Linear Graph A B C D E factor A 1　2　3　4　5　6　7 １ ① 0　0　0　0　0　0　0 E A×B 　　　② 0　0　0　1　1　1　1 ７３５　　　③ 0　1　1　0　0　1　1 　　　④ 0　1　1　1　1　0　0 ６２４　　　⑤ 1　0　1　0　1　0　1 　　　⑥ 1　0　1　1　0　1　0 D C B 　　　⑦ 1　1　0　0　1　1　0 　　　⑧ 1　1　0　1　0　0　1

3.4Construction Problem（General Case） Ex.) factorsA,B,C,D,E Special Case an OA with as few as possible of runs ・the number of factors n=5 ・strength τ=4 this can treat all L=2 order interaction effects（A×B,A×C,・・・,D×E） General Case an OA with as few as possible of runs ・the number of factorsn=5 ・　？ this can treat partial 2order interaction effects（A×B）

3.5Generator Matrix（General Case） Ex.) factorsA,B,C,D,E Special Case（A×B,A×C,・・・,D×E） any 4 columns are linearly independent A B C D E generator matrix G = General Case（A×B）・ any 4 columns are linearly independent ・any 3 columns which contain A, B are linearly independent A B C D E generator matrix G =

3.6Meaning of allocation Generator Matrix of L8 Projective Geometry （Linear Graph） 001 0　0　0　1　1　1　1 011 101 0　1　1　0　0　1　1 111 1　0　1　0　1　0　1 010 110 100

3.7Meaning of allocation Generator Matrix of L8 Projective Geometry （Linear Graph） factor A 001 A B C D E A×B 0　0　0　1　1　1　1 011 101 0　1　1　0　0　1　1 E 111 1　0　1　0　1　0　1 010 110 100 D if C, D, E are not allocated to this column, any 3 columns which contain A, B are linearly independent C B

4．Conclusion

4.1Conclusion 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.

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

An Application of Coding Theory into Experimental Design