Pang Shanqi

Shanqi Pang

Yingshan Zhang

Zhang Yingshan

Pang, Shanqi

Zhang, Yingshan

Tokyo University of Science Repository for Academic Resources / 東京理科大学学術リポジトリ

SUT Journal of MathematicsVol. 40, No. 1 (2004), 1–11Orthogonal arrays of size 108 with six-level columnsShanqi Pang and Yingshan Zhang(Received November 7, 2003)Abstract. A special structure of the two level orthogonal array of order 12is given. By using this structure and the method of constructing mixed-levelorthogonal arrays presented by Zhang, Lu and Pang (1999), a lot of mixed-level orthogonal arrays of run size 108 are obtained. Especially, we obtain anorthogonal array of size 108 with 11 6-level columns.AMS 2000 Mathematics Subject Classification. 62K15, 05B15.Key words and phrases. Mixed-level orthogonal array, Projection matrix, Per-mutation matrix.§1. IntroductionAn n×m matrix A, having ki columns with pi levels, i = 1, 2, . . . , r, m =r∑i=1ki,pi = pj for i = j, is called an orthogonal array(OA) of strength d and size nif each n× d submatrix of A contains all possible 1× d row vectors with thesame frequency. Unless stated otherwise, we use the notation Ln(pk11 · · · pkrr )for an OA of strength 2. An orthogonal array is said to have mixed-levelsif r ≥ 2. Orthogonal arrays have been used extensively in statistical designof experiments, computer science and cryptography. Constructions of mixed-level OA’s have been studied in the literature. But our knowledge aboutthe existence and nonexistence of arrays is rather limited. Hedayat, Sloaneand Stufken (1999) stated that orthogonal arrays L72(67) and L288(611) exist.How many 6-level columns can an orthogonal array of size 108 have? Thispaper presents a special structure of orthogonal array L12(211). By usingthis structure and the method of constructing mixed-level orthogonal arrayspresented by Zhang, Lu and Pang (1999), a lot of mixed-level orthogonalarrays of 108 are obtained. Especially, we obtain an orthogonal array of size108 with 11 6-level columns.12 S. PANG AND Y. ZHANG§2. Construction methodThe following deﬁnitions, notations and results are needed in the sequel.Let (r) = (0, . . . , r − 1)T , 1r be the r × 1 vector of 1’s and Ir the identityof order r. Pr = 1r1r1Tr and τr = Ir − Pr. Let ei(r) = (0 · · · 0i1 0 · · · 0)T1×r be abase vector of Rr (r-dim vector space),Nr = e1(r)eT2 (r) + · · · + er−1(r)eTr (r) + er(r)eT1 (r)andK(p, q) =p∑i=1q∑j=1ei(p)eTj (q) ⊗ ej(q)eTi (p),where ⊗ is the usual Kronecker product in the theory of matrices. Sometimes,it is necessary and easy to use the following properties of those permutationmatrices Nr and K(p, q) to obtain the orthogonal arrays needed:Nr · (r) = 1r + (r), mod rK(p, q)(1q ⊗ (p)) = (p)⊗ 1q ,K(p, q)((q) ⊗ (p)) = (p)⊗ (q).andK(p, q)(Pq ⊗ τp)KT (p, q) = τp ⊗ Pq,K(p, q)(τq ⊗ τp)KT (p, q) = τp ⊗ τq.Definition 1. Let A be an OA of strength 1, i.e.,A = (a1, . . . , am) = (S1(1r1 ⊗ (p1)), . . . , Sm(1rm ⊗ (pm))),where rjpj = n and Si is a permutation matrix, for i = 1, . . . ,m. Then theprojection matrix Sj(Prj ⊗ τpj )STj is called the matrix image (MI) of the jthcolumn aj of A, denoted by m(aj). The MI of a subarray of A is deﬁned asthe sum of the MI’s of all its columns. In particular, we denote the MI of Aby m(A).From the deﬁnition of matrix image, we havem(1r) = Pr,m((r)) = τr.Definition 2. Suppose that Ln is an OA with entries from a ﬁnite addi-tive group G, and that a and b are two columns of Ln. i.e., a = Ln(p) =(a1, . . . , an)T and b = Ln(q) = (b1, . . . , bn)T . The generalized Hadamardproduct of a and b, denoted a b, is deﬁned asa b = (h(a1, b1), . . . , h(an, bn))T = (a1q + b1, . . . , anq + bn)T = Ln(t),where t = pq and h(ai, bi) = aiq + bi for i = 1, . . . , n.ORTHOGONAL ARRAYS OF SIZE 108 WITH SIX-LEVEL COLUMNS 3Deﬁnitions 1 and 2 can be found in Zhang, et al (2001).Lemma 1. For any permutation matrix S and any array L , m(S(L⊗1r)) =S(m(L)⊗ Pr)ST , and m(S(1r ⊗ L)) = S(Pr ⊗m(L))ST .Lemma 2. Let A be an orthogonal array of strength 1, i.e.,A = (a1, . . . , am) = (S1(1r1 ⊗ (p1)), . . . , Sm(1rm ⊗ (pm))),where rjpj = n and Si is a permutation matrix, for i = 1, . . . ,m. The follow-ing statements are equivalent.(1) A is an orthogonal array of strength 2.(2) m(A) is a projection matrix.(3) m(ai)m(aj) = 0 (i = j).(4) The projection matrix τn can be decomposed as τn = m(a1)+ · · ·+m(am)+∆, where rk(∆) = n− 1−m∑j=1(pj − 1) is the rank of the matrix .Lemma 3. Suppose τn1 =∑j SjASTj and τn2 =∑j TjBTTj are orthogonaldecompositions of τn1 and τn2, respectively, where the Sj ’s and Tj’s are per-mutation matrices and n = n1n2. Then τn1n2 can be orthogonally decomposedintoτn1n2 =∑j(Sj ⊗ Tj)(A⊗ Pn2 + In1 ⊗ B)(STj ⊗ T Tj ).If there exists an orthogonal array H such that m(H) ≤ In1 ⊗ B + A ⊗ Pn2,thenL = ((S1 ⊗ T1)H, (S2 ⊗ T2)H, . . . )is also an orthogonal array.Lemmas 1,2 and 3 can be found in Zhang, Lu and Pang (1999).§3. A special structure of orthogonal array L12(211)Consider the following orthogonal array L12(211)L12(211) = [a1, . . . , a11] =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝000000001110000011100000111000000010110111100110110110101110110011101100111011010111011010011101011111000010101101010010011001010001⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.4 S. PANG AND Y. ZHANGLet A1 = (a3, a4, a5), A2 = (a6, a7, a8) and A3 = (a9, a10, a11). It is easyto see that A2 = TA1 and A3 = QA1, where T = diag(T1, T2, T3, T4), Q =diag(Q1, Q2, Q3, Q4),T1 =⎛⎝1 0 00 0 10 1 0⎞⎠ , T2 = I3, T3 = T4 =⎛⎝0 0 11 0 00 1 0⎞⎠andQ1 = Q2 =⎛⎝0 0 10 1 01 0 0⎞⎠ , Q3 = Q4 =⎛⎝0 1 01 0 00 0 1⎞⎠ .From the deﬁnition of orthogonal array, there exists a permutation matrix Ssuch thatS(a3, a4) = (16 ⊗ (2),13 ⊗ ((2)⊕ (2)))where (2) ⊕ (2) = (0, 1, 1, 0)T , where ⊕ is the usual Kronecker sum in thetheory of matrix. LetS =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝100000000000001000000000000000000010000010000000010000000000000001000000000000001000000000010000000000000001000000100000000100000000000000000100⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠and L112(211) = S(a1, a2, . . . , a11) = (b1, b2, . . . , b11), then we have b1 =(001000111101)T , b2 = (001101001011)T , b3 = 16 ⊗ (2), b4 = 13 ⊗ ((2) ⊕ (2))and b5 = (010100111010)T . Let B1 = (b3, b4, b5), B2 = (b6, b7, b8) andB3 = (b9, b10, b11), then B2 = STS−1B1 and B3 = SQS−1B1. It follows thatL112(211) = (M1B1,M2B1,M3B1, b1, b2), where M1 = I12,M2 = STS−1,M3 =ORTHOGONAL ARRAYS OF SIZE 108 WITH SIX-LEVEL COLUMNS 5SQS−1. If setM4 =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝100000000000000010000000010000000000001000000000000000001000000000100000000100000000000000010000000001000000000000000001000000000010000000000100⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,thenL112(211) = (M1B1,M2B1,M3B1,M4(b3, b4)).L112(29) = (b5,M2B1,M3B1,M4(b3, b4)).L112(28) = (M2B1,M3B1,M4(b3, b4)).The structure of orthogonal array L12(211) is useful for construction ofother arrays.We ﬁrst give some smaller orthogonal arrays and their matrix images whichwill be used.Theorem 1. There exist orthogonal arrays L36(63) and L36(6234) such thatm(L36(63)) ≤ I12⊗ τ3 +m(B1)⊗P3 and m(L36(6234)) ≤ I12⊗ τ3 + (P6⊗ τ2 +P3 ⊗ τ2 ⊗ τ2)⊗ P3.Proof. An orthogonal array L36(63 ·28) can be obtained by Zhang, et al (2001)from the following formula:L36(63 · 28) = [13 ⊗ L12(28), (((3) ⊕ (3))⊗ 14) (118 ⊗ (2)),(((3)⊕ (3)+2) ⊗ 14) (19 ⊗ ((2)⊕ (2))), ((3) ⊕ a) (13 ⊗ b)],where the operation is deﬁned in Deﬁnition 2 and OA L12(28) satisﬁes thecondition:L112(211) = [16 ⊗ (2),13 ⊗ ((2)⊕ (2)), b5, L12(28)] = [B1, L12(28)].Let L136(63 · 28) = K(12,3)L36(63 · 28), then L136(63 · 28) contains a subarrayL12(28)⊗ 13. On the other hand, we havem(L136(63 ·28)) ≤ τ36 = I12⊗τ3+τ12⊗P3 = I12⊗τ3+(m(B1)+m(L12(28)))⊗P3.Deleting L12(28)⊗ 13 from L136(63 · 28), we get an OA L36(63) satisfyingm(L36(63)) ≤ I12 ⊗ τ3 + m(B1)⊗ P3.6 S. PANG AND Y. ZHANGSimilarly and from the OA L36(62 ·34 ·29) in Zhang et al(2001), one can obtainan OA L36(6234) such that m(L36(6234)) ≤ I12⊗τ3+(P6⊗τ2+P3⊗τ2⊗τ2)⊗P3.This completes the proof.Corollary 1. There are orthogonal arrays L36(31219), L36(32212), L36(3325),L36 (6131210), L36(613223), L36(623121), L36(613822), L36(623421) andL36(31223) whose matrix images are less than or equal to I12⊗τ3+m(B1)⊗P3.There exist orthogonal arrays L36(31219), L36(32211), L36(3324), L36(613129),L36(613222), L36(613821) and L36(31222) whose matrix images are less than orequal to I12 ⊗ τ3 + (P6 ⊗ τ2 + P3 ⊗ τ2 ⊗ τ2)⊗ P3.Proof. The proof is similar to that of Theorem 1. Using those orthogonalarrays in Zhang, et al (1999) and Zhang, et al (2001), we can obtain theorthogonal arrays we needed.Now we illustrate construction of orthogonal array L108(61134).Theorem 2. There exist orthogonal arrays L108(61134), L108(6103821), etc.Proof. There exists Sj (j = 1, 2, 3, 4) such thatτ9 =4∑j=1Sj(τ3 ⊗ P3)STj ,where S1 = I9, S2 = K(3, 3), S3 = K(3, 3)diag(I3, N3, N23 )K(3, 3)T , andS4 = K(3, 3) diag(I3, N23 , N3) K(3, 3)T . And from the OA L112(211), we havethe following decomposition of the projection matrix τ12:τ12 =3∑j=1(Mjm(B1)MTj ) + M4(P6 ⊗ τ2 + P3 ⊗ τ2 ⊗ τ2)MT4 .Hence projection matrix τ108 can be orthogonally decomposed asτ108 = I12 ⊗ τ9 + τ12 ⊗ P9=3∑j=1(Mj ⊗ Sj)(I12 ⊗ τ3 ⊗ P3 + m(B1)⊗ P9)(Mj ⊗ Sj)T+ (M4 ⊗ S4)(I12 ⊗ τ3 ⊗ P3 + (P6 ⊗ τ2 + P3 ⊗ τ2 ⊗ τ2) ⊗ P9)(M4 ⊗ S4)T .From Lemma 3, we obtain an orthogonal array L108(61134) as follows(see Table1):L108(61134) = [(M1 ⊗ S1)(L36(63)⊗ 13),(M2 ⊗S2)(L36(63)⊗ 13), (M3 ⊗ S3)(L36(63)⊗ 13), (M4 ⊗ S4)(L36(6234)⊗ 13)].Using the arrays in Corollary 1 to replace the arrays L36(63) and L36(6234)in L108(61134) respectively, we can construct 1760 (220× 8) orthogonal arrayssuch as L108(6103821), L108(6103222), L108(6103129), etc.ORTHOGONAL ARRAYS OF SIZE 108 WITH SIX-LEVEL COLUMNS 7AcknowledgementsThis work is supported by NSF of the Education Department of Henan Province,China (20011100003), University Backbone Teachers Foundation of the Edu-cation Department of Henan Province, and NSF of Henan Province, China(0411011100).8 S. PANG AND Y. ZHANGTable 1. The orthogonal array L108(61134) = s1 − s11, t1 − t4No. s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 t1 − t41 0 0 2 0 0 2 1 1 3 0 0 11222 0 0 2 2 2 4 3 3 5 4 4 00113 0 0 2 4 4 0 5 5 1 2 2 22004 2 2 4 0 0 2 3 3 5 2 2 22005 2 2 4 2 2 4 5 5 1 0 0 11226 2 2 4 4 4 0 1 1 3 4 4 00117 4 4 0 0 0 2 5 5 1 4 4 00118 4 4 0 2 2 4 1 1 3 2 2 22009 4 4 0 4 4 0 3 3 5 0 0 112210 1 1 3 2 4 2 0 0 2 2 4 121211 1 1 3 4 0 4 2 2 4 0 2 010112 1 1 3 0 2 0 4 4 0 4 0 202013 3 3 5 2 4 2 2 2 4 4 0 202014 3 3 5 4 0 4 4 4 0 2 4 121215 3 3 5 0 2 0 0 0 2 0 2 010116 5 5 1 2 4 2 4 4 0 0 2 010117 5 5 1 4 0 4 0 0 2 4 0 202018 5 5 1 0 2 0 2 2 4 2 4 121219 0 1 4 5 2 2 5 2 2 1 1 112220 0 1 4 1 4 4 1 4 4 5 5 001121 0 1 4 3 0 0 3 0 0 3 3 220022 2 3 0 5 2 2 1 4 4 3 3 220023 2 3 0 1 4 4 3 0 0 1 1 112224 2 3 0 3 0 0 5 2 2 5 5 001125 4 5 2 5 2 2 3 0 0 5 5 001126 4 5 2 1 4 4 5 2 2 3 3 220027 4 5 2 3 0 0 1 4 4 1 1 112228 1 0 1 1 0 1 1 0 1 0 1 221129 1 0 1 3 2 3 3 2 3 4 5 110030 1 0 1 5 4 5 5 4 5 2 3 002231 3 2 3 1 0 1 3 2 3 2 3 002232 3 2 3 3 2 3 5 4 5 0 1 221133 3 2 3 5 4 5 1 0 1 4 5 110034 5 4 5 1 0 1 5 4 5 4 5 110035 5 4 5 3 2 3 1 0 1 2 3 002236 5 4 5 5 4 5 3 2 3 0 1 221137 2 4 2 1 1 3 2 4 2 4 2 122138 2 4 2 3 3 5 4 0 4 2 0 011039 2 4 2 5 5 1 0 2 0 0 4 200240 4 0 4 1 1 3 4 0 4 0 4 2002ORTHOGONAL ARRAYS OF SIZE 108 WITH SIX-LEVEL COLUMNS 941 4 0 4 3 3 5 0 2 0 4 2 122142 4 0 4 5 5 1 2 4 2 2 0 011043 0 2 0 1 1 3 0 2 0 2 0 011044 0 2 0 3 3 5 2 4 2 0 4 200245 0 2 0 5 5 1 4 0 4 4 2 122146 3 5 4 3 5 4 4 3 3 2 5 212147 3 5 4 5 1 0 0 5 5 0 3 101048 3 5 4 1 3 2 2 1 1 4 1 020249 5 1 0 3 5 4 0 5 5 4 1 020250 5 1 0 5 1 0 2 1 1 2 5 212151 5 1 0 1 3 2 4 3 3 0 3 101052 1 3 2 3 5 4 2 1 1 0 3 101053 1 3 2 5 1 0 4 3 3 4 1 020254 1 3 2 1 3 2 0 5 5 2 5 212155 2 5 3 3 4 1 2 5 3 1 0 221156 2 5 3 5 0 3 4 1 5 5 4 110057 2 5 3 1 2 5 0 3 1 3 2 002258 4 1 5 3 4 1 4 1 5 3 2 002259 4 1 5 5 0 3 0 3 1 1 0 221160 4 1 5 1 2 5 2 5 3 5 4 110061 0 3 1 3 4 1 0 3 1 5 4 110062 0 3 1 5 0 3 2 5 3 3 2 002263 0 3 1 1 2 5 4 1 5 1 0 221164 3 4 1 5 3 4 5 3 4 3 4 212165 3 4 1 1 5 0 1 5 0 1 2 101066 3 4 1 3 1 2 3 1 2 5 0 020267 5 0 3 5 3 4 1 5 0 5 0 020268 5 0 3 1 5 0 3 1 2 3 4 212169 5 0 3 3 1 2 5 3 4 1 2 101070 1 2 5 5 3 4 3 1 2 1 2 101071 1 2 5 1 5 0 5 3 4 5 0 020272 1 2 5 3 1 2 1 5 0 3 4 212173 4 2 1 0 1 4 4 2 1 3 5 121274 4 2 1 2 3 0 0 4 3 1 3 010175 4 2 1 4 5 2 2 0 5 5 1 202076 0 4 3 0 1 4 0 4 3 5 1 202077 0 4 3 2 3 0 2 0 5 3 5 121278 0 4 3 4 5 2 4 2 1 1 3 010179 2 0 5 0 1 4 2 0 5 1 3 010180 2 0 5 2 3 0 4 2 1 5 1 202010 S. PANG AND Y. ZHANG81 2 0 5 4 5 2 0 4 3 3 5 121282 5 3 4 2 5 3 3 4 1 5 2 211283 5 3 4 4 1 5 5 0 3 3 0 100184 5 3 4 0 3 1 1 2 5 1 4 022085 1 5 0 2 5 3 5 0 3 1 4 022086 1 5 0 4 1 5 1 2 5 5 2 211287 1 5 0 0 3 1 3 4 1 3 0 100188 3 1 2 2 5 3 1 2 5 3 0 100189 3 1 2 4 1 5 3 4 1 1 4 022090 3 1 2 0 3 1 5 0 3 5 2 211291 4 3 3 4 3 3 3 5 4 4 3 211292 4 3 3 0 5 5 5 1 0 2 1 100193 4 3 3 2 1 1 1 3 2 0 5 022094 0 5 5 4 3 3 5 1 0 0 5 022095 0 5 5 0 5 5 1 3 2 4 3 211296 0 5 5 2 1 1 3 5 4 2 1 100197 2 1 1 4 3 3 1 3 2 2 1 100198 2 1 1 0 5 5 3 5 4 0 5 022099 2 1 1 2 1 1 5 1 0 4 3 2112100 5 2 2 4 2 1 0 1 4 5 3 1221101 5 2 2 0 4 3 2 3 0 3 1 0110102 5 2 2 2 0 5 4 5 2 1 5 2002103 1 4 4 4 2 1 2 3 0 1 5 2002104 1 4 4 0 4 3 4 5 2 5 3 1221105 1 4 4 2 0 5 0 1 4 3 1 0110106 3 0 0 4 2 1 4 5 2 3 1 0110107 3 0 0 0 4 3 0 1 4 1 5 2002108 3 0 0 2 0 5 2 3 0 5 3 1221References[1] Hedayat, A.S., Sloane, N.J.A. and Stufken, J.(1999), Orthogonal arrays: Theoryand Applications, Springer, New York.[2] Zhang, Y.S., Lu, Y.Q. and Pang, S.Q.(1999), Orthogonal arrays obtained byorthogonal decomposition of projection matrices. Statist. Sinica. 2, 595-604.[3] Zhang, Y.S., Pang, S.Q. and Wang, Y.P.(2001), Orthogonal arrays obtained bythe generalized Hadamard product. Discrete Math. 238, 151-170.Shanqi PangDepartment of Mathematics, Henan Normal UniversityXinxiang, 453002E-mail : pangshanqi@263.netORTHOGONAL ARRAYS OF SIZE 108 WITH SIX-LEVEL COLUMNS 11Yingshan ZhangDepartment of Computer Science, Henan Normal UniversityXinxiang, 453002

Orthogonal arrays of size 108 with six-level columns

https://tus.repo.nii.ac.jp/?action=repository_action_common_download&item_id=1207&item_no=1&attribute_id=22&file_no=1

Orthogonal arrays of size 108 with six-level columns

Abstract

Similar works

Full text

Available Versions

Tokyo University of Science Repository for Academic Resources / 東京理科大学学術リポジトリ