1. 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 Experimental Design Shigeichi Hirasawa Department of Industrial and Management Systems Engineering, School of Science and Engineering, Waseda University - Construction Methods for Unequal Orthogonal Arrays -

2. No.1 １． Introduction 序論

3. No.2 1.1 Abstract Orthogonal Arrays (OAs) Error-Correcting Codes (ECCs) Experimental DesignCoding Theory ・ relations between OAs and ECCs ・ the table of OAs and Hamming codes ・ the table of OAs + allocation table of OA L 8 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

4. No.3 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

5. No.4 準備 2 ． Preliminary

6. No.5 実験計画法 Experimental Design

7. No.6 ２．１ Experimental Design ( 実験計画 法 ) ・ Factor A (materials) A 0 （ A company ）， A 1 （ B company ） ・ Factor B （ machines ） B 0 （ new ）， B 1 （ old ） ・ Factor C （ temperatures ） C 0 （ 100 ℃）， C 1 （ 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

8. No.7 Complete Array Ａ Ｂ Ｃ ０ 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 A 0,B 0,C 0 experiments with all combination of levels 完全配列 実験

9. No.8 Ａ Ｂ Ｃ ０ ０ ０ ０ １ １ １ ０ １ １ １ ０ ② ③ ④ Orthogonal Array (OA) : OA(M, n, s,τ) (s=2) 100110 000010 011 111 101 001 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

10. No.9 Ａ Ｂ Ｃ ０ ０ ０ ０ １ １ １ ０ １ １ １ ０ ② ③ ④ 100110 000010 011 111 101 001 Experiment ① Orthogonal Array (OA) : OA(M, n, s,τ) (s=2) subset (subspace) of complete array 直交配列 部分空間 strength τ=2 every 2 columns contains each 2- tuple exactly same times as row 強さ

11. No.10 Ａ Ｂ Ｃ ０ ０ ０ ０ １ １ １ ０ １ １ １ ０ ② ③ ④ 100110 000010 011 111 101 001 Experiment ① Orthogonal Array (OA) : OA(M, n, s,τ) (s=2) subset (subspace) of complete array 直交配列 部分空間 strength τ=2 every 2 columns contains each 2- tuple exactly same times as row 強さ

12. No.11 Ａ Ｂ Ｃ ０ ０ ０ ０ １ １ １ ０ １ １ １ ０ ② ③ ④ 100110 000010 011 111 101 001 Experiment ① Orthogonal Array (OA) : OA(M, n, s,τ) (s=2) subset (subspace) of complete array 直交配列 部分空間 strength τ=2 every 2 columns contains each 2- tuple exactly same times as row 強さ

13. No.12 Parameters of OAs ・ the number of factors n ・ the number of runs M ・ strength τ=2t A B C ① ② ③ ④ ０ ０ ０ ０ １ １ １ ０ １ １ 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

14. No.13 Generator Matrix of an OA : G Ex.) orthogonal array ｛ 000, 011, 101, 110 ｝ （ ○ ， ○ ， ○ ） = （ □ ， □ ） ０ １ １ １ ０ １ OA each k-tuple (k=2) based on ｛ 0,1 ｝ 2 2 k =M generator matrix G ABC ABC 2.1.3 Generator Matrix ( 生成行 列 ) To construct OAs is to construct generator matrix

15. No.14 orthogonal array ｛ 000, 011, 101, 110 ｝ （ ０， ０， ０ ） = （ ０，０ ） ０ １ １ ABC ABC １ ０ １ OA generator matrix G Ex.) To construct OAs is to construct generator matrix Generator Matrix of an OA : G 2.1.3 Generator Matrix ( 生成行 列 ) each k-tuple (k=2) based on ｛ 0,1 ｝ 2 2 k =M

16. No.15 orthogonal array ｛ 000, 011, 101, 110 ｝ （ ０， １， １ ） = （ １，０ ） ０ １ １ ABC ABC １ ０ １ OA To construct OAs is to construct generator matrix generator matrix G Ex.) Generator Matrix of an OA : G 2.1.3 Generator Matrix ( 生成行 列 ) each k-tuple (k=2) based on ｛ 0,1 ｝ 2 2 k =M

17. No.16 orthogonal array ｛ 000, 011, 101, 110 ｝ （ １， ０， １ ） = （ ０，１ ） ０ １ １ ABC ABC １ ０ １ OA generator matrix G Ex.) To construct OAs is to construct generator matrix Generator Matrix of an OA : G 2.1.3 Generator Matrix ( 生成行 列 ) each k-tuple (k=2) based on ｛ 0,1 ｝ 2 2 k =M

18. No.17 orthogonal array ｛ 000, 011, 101, 110 ｝ （ １， １， ０ ） = （ １，１ ） ０ １ １ ABC ABC １ ０ １ OA generator matrix G Ex.) To construct OAs is to construct generator matrix Generator Matrix of an OA : G 2.1.3 Generator Matrix ( 生成行 列 ) each k-tuple (k=2) based on ｛ 0,1 ｝ 2 2 k =M

19. No.18 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 １ １ ０ １ G = 3 2 the number of factors n=3 the number of runs M=2 2 any 2 columns are linearly independent strength τ=2 Ex.)

20. No.19 Parameters of OAs and Generator Matrix orthogonal arrays ｛ 000, 011, 101, 110 ｝ ０ １ １ １ ０ １ G = 3 2 any 2 columns are linearly independent Ex.) 0 1 1 0 + 0 0 ≠ ・ the number of factors n=3 ・ the number of runs M=4 ・ strength τ=2 the number of factors n=3 the number of runs M=2 2 strength τ=2

21. No.20 Parameters of OAs and Generator Matrix orthogonal arrays ｛ 000, 011, 101, 110 ｝ ０ １ １ １ ０ １ G = 3 2 any 2 columns are linearly independent Ex.) 0 1 1 1 + 0 0 ≠ ・ the number of factors n=3 ・ the number of runs M=4 ・ strength τ=2 the number of factors n=3 the number of runs M=2 2 strength τ=2

22. No.21 Parameters of OAs and Generator Matrix orthogonal arrays ｛ 000, 011, 101, 110 ｝ ０ １ １ １ ０ １ G = 3 2 any 2 columns are linearly independent Ex.) 1 0 1 1 + 0 0 ≠ ・ the number of factors n=3 ・ the number of runs M=4 ・ strength τ=2 the number of factors n=3 the number of runs M=2 2 strength τ=2

23. No.22 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=2 m ・ 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

24. No.23 Coding Theory

25. No.24 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 0001000 Ex.) encoder channel decoder noise codewords 符号語

26. No.25 Error-Correcting Codes subspace of linear vector space 100110 000010 011 111 101 001 ０ １ ００ ０ １１ １ codeword Ex.) 誤り訂正符号 部分空間 符号語

27. No.26 ・ code length n ・ the number of information symbols k ・ minimum distance d=2t + 1 this can correct t errors trade off ０００ ０ １１１ １ 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) code Parameters 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

28. No.27 Parity Check Matrix of ECCs Ex.) (3,1,3) code ｛ 000, 111 ｝ parity check matrix H = 0 1 1 1 0 1 000000 = 0 111111 = 0 codeword To construct of linear codes is to construct parity check matrix 2.2.3 Parity Check Matrix HxT=0HxT=0

29. No.28 Parameters of ECCs and Parity Check Matrix ・ code length n=3 ・ the number of information symbols k=1 ・ minimum distance d=3 ０ １ １ １ 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 ｝

30. No.29 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=2 m ・ strength τ=2t ・ code length n ・ the number of information symbols n-m ・ minimum distance d=2t + 1 this can correct all t errors any d-1=2t columns are linearly independent this can treat all t order interaction effect

31. No.30 3 ． Relation Between OAs and ECCs 関係

32. No.31 3.1 OAs and ECCs ０ １ １ １ ０ １ G = 100110 000010 011 111 101 001 100110 000010 011 111 101 001 OA with generator matrix G ECC with parity check matrix G

33. No.32 OAs and ECCs [HSS ‘99] G = n m OAs with generator matrix G ・ the number of factors n ・ the number of runs M=2 m ・ 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 2t columns are linearly independent this can treat all t order interaction effect ECCs with parity check matrix G

34. No.33 Table of OAs and Hamming Codes 直交表

35. No.34 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 １ ０ １ １ ０ ０ １ １ １ ０ １ ０ １ ０ １ 0 0 1 1 1 1 ・・・・・・ G = 3 n=7

36. No.35 3.2 Matrix in which any 2 columns are linearly ０ ０ ０ １ １ １ １ ０ １ １ ０ ０ １ １ １ ０ １ ０ １ ０ １ G = 3 independent ① 0 0 1 0 1 0 +≠ 0 0 0 an OA with strength τ=2 ， a linear code with minimum distance 0 0 1 1 1 1 ・・・・・・ n=7

37. No.36 3.2 Matrix in which any 2 columns are linearly an OA with strength 2 ， a linear code with minimum distance ０ ０ ０ １ １ １ １ ０ １ １ ０ ０ １ １ １ ０ １ ０ １ ０ １ G = 3 independent ② ・ table of OA L 8 ・（ 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 0 0 1 1 1 1 ・・・・・・ n=7

38. No.37 3.2 Matrix in which any 2 columns are linearly an OA with strength 2 ， a linear code with minimum distance independent ① ・ table of OA L 16 ・（ 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 １ ０ １ １ ０ ０ １ １ ０ ０ １ １ ０ ０ １ １ １ 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

39. No.38 Table of OAs + allocation 直交表割付

40. No.39 1 2 3 4 5 6 7 ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ 3.3 Example （ Allocation to L 8 ） 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 10 0 0 1 1 1 1 0 1 1 0 0 1 10 1 1 0 0 1 1 0 1 1 1 1 0 00 1 1 1 1 0 0 1 0 1 0 1 0 11 0 1 0 1 0 1 1 0 1 1 0 1 01 0 1 1 0 1 0 1 1 0 0 1 1 01 1 0 0 1 1 0 1 1 0 1 0 0 11 1 0 1 0 0 1 L 8 Linear Graph １ ２４ ３５ ６ ７ 線点図

41. No.40 1 2 3 4 5 6 7 ② ③ ④ ⑤ ⑥ ⑦ ⑧ 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 1 10 0 0 1 1 1 1 0 1 1 0 0 1 10 1 1 0 0 1 1 0 1 1 1 1 0 00 1 1 1 1 0 0 1 0 1 0 1 0 11 0 1 0 1 0 1 1 0 1 1 0 1 01 0 1 1 0 1 0 1 1 0 0 1 1 01 1 0 0 1 1 0 1 1 0 1 0 0 11 1 0 1 0 0 1 １ ２４ ３５ ６ ７ factor A B D E A×BA×B BADEC C 3.3 Example （ Allocation to L 8 ） L 8 Linear Graph ① 線点図

42. No.41 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

43. No.42 3.5 Generator Matrix （ General Case ） Special Case （ A×B,A×C, ・・・,D×E ） General Case （ A×B ） ABCDE generator matrix G = any 4 columns are linearly independent ABCDE ・ 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 =

44. No.43 3.6 Meaning of allocation Generator Matrix of L 8 Projective Geometry （ Linear Graph ） 0 0 0 1 1 1 10 0 0 1 1 1 1 0 1 1 0 0 1 10 1 1 0 0 1 1 1 0 1 0 1 0 11 0 1 0 1 0 1 001 010100 011 101 110 111

45. No.44 0 0 0 1 1 1 10 0 0 1 1 1 1 0 1 1 0 0 1 10 1 1 0 0 1 1 1 0 1 0 1 0 11 0 1 0 1 0 1 001 010100 011 101 110 BADEC factor A B D C E if C, D, E are not allocated to this column, any 3 columns which contain A, B are linearly independent A×BA×B 111 3.7 Meaning of allocation Generator Matrix of L 8 Projective Geometry （ Linear Graph ）

46. No.45 4 ． Unequal Error Protection Codes and OAs

47. No.46 4.1 Unequal Error Protection Codes Unequal Error Protection Codes ( ○, ○, ○ ) codeword error protection levels are equal in each position of a codeword ( ○, ○, ○ ) codeword t+1tt error protection level t in each position error protection level (t 1,t 2,t 3 ) = (t+1, t, t) ttt Error-Correcting Codes Unequal Error Protection Codes error protection levels are unequal in each position of a codeword → minimum distance d=2t +1 → separation d i =2t i +1 this is used to send numerical data

48. No.47 Construction Problem of Unequal Error Protection Codes Error-Correcting Codes 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 (d 1,d 2, ・・・,d n )

49. No.48 Unequal Error Protection Codes and Parity Check Matrix Error-Correcting Codes Unequal Error Protection Codes H=1in ･･ ･ minimum distance d H=1in any d i -1 columns that contain i-th column are linearly independent separation (d 1, ･･･, d i, ･･･, d n ) ･･ ･ any d-1 columns are linearly independent

50. No.49 4.2 Classification of OA ① OA （ General Case ） ② OA with unequal strength (τ 1, τ 2, ・・・, τ n ) → this can treat all τ i /2 = t i -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

51. No.50 4.2 Classification of OA ① OA （ General Case ） ② OA with unequal strength (τ 1, τ 2, ・・・, τ n ) Unequal Protection Codes ③ OA with (equal) strength τ Error-Correcting Codes ① ③ ② Unequal Protection Codes Error-Correcting Codes

52. No.51 5. Examples of OAs with unequal strength

53. No.52 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 2 18 ・ strength 6 ・ number of factors 63 ・ number of experiments 2 12 ・ 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

54. No.53 6 ． Conclusion

55. No.54 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.

56. No.55 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