Construction of Ternary Orthogonal Arrays by Kronecker Sum KISHORE SINHA1, *SOMESH KUMAR2 AND ANINDITA SEN GUPTA2

Summary Kronecker sum of ternary orthogonal arrays has been introduced. Kronecker sum of a ternary orthogonal array obtained from a BIB design and another ternary orthogonal array is described. Some new symmetrical ternary orthogonal arrays not found in the well known tables are obtained using this technique. Keywords: Kronecker sum, orthogonal array, BIB design. MSC: Primary : 05B15; Secondary : 62K15

1.

INTRODUCTION

The Kronecker sum of orthogonal arrays has been investigated by Shrikhande (1964), Wang and Wu (1991), Zhang, Weiguo, Mao and Zheng (2006) and Sinha, Vellaisamy and Sinha (2008). Balanced arrays of strength t, denoted by BA(b, m, s, t) {µ x1 . . . xt}, is an m × b matrix B with elements belonging to a set {0, 1, . . . , s − 1} of s symbols, m constraints (factors), b assemblies (runs), and strength t, such that every t × b submatrix of B contains the ordered t × 1 column vector (x1, . . . , xt)′, µ x1...xt times, where µ x1...xt is invariant under any permutation of x1, . . . , xt. When µ x1...xt is a constant, λ (say), for all x1, . . . , xt, the BA is called an orthogonal array, denoted by OA(b, m, s, t) with index λ. Here b = λst. When each of k factors have equal number of symbols (levels), say, s, the orthogonal array is called symmetrical. An orthogonal array with s = 3 is called ternary orthogonal array. One can refer to Hedayat et al. (1999) for a comprehensive treatise on orthogonal arrays. Following Sinha et al. (1979), Kronecker sum of matrices A of dimension m × n and B of dimension p × q, may be defined as A / B = A 1 J + J 1 B, where 1 denotes the usual J. Stat. & Appl. Vol. 4, No. 2-3, 2009, pages 475-478 © MD Publications Pvt Ltd Corresponding Author Email:[email protected], [email protected]

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Kishore Sinha, Somesh Kumar and Anindita Sen Gupta

Kronecker product and J is a matrix with all its elements unity but is of dimension p×q in the first term and m × n in the second term. Sinha et al. (2008) defined Kronecker sum of binary orthogonal arrays. Here we consider Kronecker sum of ternary orthogonal arrays A and B which is equivalent to replacing ith element in A by (i + B) mod 3; i = 0, 1, 2; that is, i is added mod 3 to each element of B. In this paper, the Kronecker sum of a ternary orthogonal array obtained from a BIB design and a ternary orthogonal array is described. We obtain symmetrical ternary orthogonal arrays utilizing this technique. Some of these orthogonal arrays are not found in table 12.6(e) of ternary orthogonal arrays by Hedayat et al. (1999) or in the web-pages maintained by Warren, F.Kuhfeld.

2.

THE CONSTRUCTION

Lemma 2.1: The existence of a BIB design with parameters: v′, b′ = 3(r′ − λ′), r′, k′, λ′ implies the existence of an OA(9(r′ − v′), v′, 3, 2) with index µ = r′ − λ′. Proof: Let A0 be the incidence matrix of the BIB design (v′, b′, r′, k′, λ′). It is well known that the incidence matrix of a BIB design is a balanced array (b′, v′, 2, 2) with indices: = b′ − 2r′ + λ′, = r′ − λ′. Next we obtain A1 and A2 by adding 1 and 2, mod 3, respectively, to each element of A0. Further, the juxtaposition of the arrays: [A0 : A1 : A2] ′ gives a BA(3b′ , v′ , 3 , 2) with indices = λ′ + (b′ − 2r′ + λ′) = b′ − 2 (r′ − λ′) and = r ′− λ′ ; i , j = 0, 1, 2 and i ≠ j. When the BIB design satisfies b′ = 3(r′ − λ′), we get an OA(9(r′ − λ′), v′, 3, 2) with index µ = r′ − λ′. Corollary 2.1: Corresponding to the BIB designs v = 3 = b, r = 2 = k, λ = 1 and v = 4, b = 6, r = 3, k = 2, λ = 1, we get respective orthogonal arrays: OA(9, 3, 3, 2), µ = 1 and OA(18, 4, 3, 2), µ = 2. Theorem 2.1: The Kronecker sum of a BIB design with parameters: v′, b′ = 3(r′ − λ′), r′, k′, λ′ and an OA(N, k, 3, t) with index µ implies the existence of OA(b′N, v′k, 3, 2) with index µ = (r′ − λ′) N/3. Proof: Let A0 be the incidence matrix of the BIB design: v′, b′ = 3(r′ − λ′), r′, k′, λ′ and let B denote an OA(N, k, 3, t) with index µ. Let us consider the Kronecker sum B / A0, which is equivalent to replacing a symbol i; (i = 0, 1, 2) in orthogonal array B by Ai. As it is known from the above lemma that [A0 : A1 : A2]′ forms an OA(9(r′ − λ′), v′, 3, 2) and B is also an orthogonal array, we get an OA(b′ N, v′ k, 3, 2) with index µ = (r′ − λ′) N/3 by the Kronecker sum. Journal of Statistics & Applications

Construction of Ternary Orthogonal Arrays by Kronecker Sum 477

Example: Given below is a ternary OA(27, 12, 3, 2) (transposed) obtained from BIB design v = 3 = b, r = 2 = k, λ = 1 and OA(9, 4, 3, 2).

Remark 2.1: An easy generalization to the above method for s ≥ 4 does not seem feasible. 3.

TABLE OF ORTHOGONAL ARRAYS

We give below a table of ternary (s = 3) orthogonal arrays of strength two (t = 2) obtained using Theorem 2.1. Here N and k stand for numbers of rows and columns respectively. The OA no.s 5, 8,9,10 are not found in Hedayat et al. (1999) or the web-pages: http : //support.sas.com/techsup/technote/ts723.html Table 3.1: Ternary orthogonal arrays (N < 1000, k < 100) of strength 2

Journal of Statistics & Applications

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Kishore Sinha, Somesh Kumar and Anindita Sen Gupta

maintained by Warren, F. Kuhfeld. The OA no.4 for 135 runs is not found in Hedayat et al. (1999) and is better than the ones reported in the web-pages. For other orthogonal arrays, although the corresponding OAs with same runs but more constraints are available in Hedayat et al. (1999) or the web-pages, but our OAs give alternative procedure of construction with possibly different properties. ACKNOWLEDGEMENT The authors are highly thankful to the referees for helpful suggestions. References 1. 2. 3. 4. 5. 6.

Hedayat, A., Sloane, N. J. A. and Stufken, J. (1999). Orthogonal Arrays. Springer-Verlag, New York. Shrikhande, S. S. (1964). Generalized Hadamard matrices and orthogonal arrays of strength two. Canadian J. Math., 16, 736-740. Sinha, K., Mathur, S. N. and Nigam, A. K. (1979). Kronecker sum of incomplete block designs. Utilitas Mathematica, 16, 157-164. Sinha, K., Vellaisamy, P., Sinha, N. (2008). Kronecker sum of binary orthogonal arrays. Utilitas Mathematica, 75, 249-257. Wang, J. C. and Wu, C. F. J. (1991). An approach to construction of asymmetrical orthogonal arrays. J. Amer. Statist. Assoc., 86, 450-456. Zhang, Y., Weiguo, L., Mao, S., Zheng, Z. (2006). A simple method for constructing orthogonal arrays by the Kronecker sum. J. Syst. Sci. Complexity, 19, 266-273.

ABOUT THE AUTHORS: Kishore Sinha Department of Statistics, Birsa Agricultural University, Ranchi - 834006, India Somesh Kumar Department of Mathematics, Indian Institute of Technology, Kharagpur-721 302, India Anindita Sen Gupta Department of Mathematics, Indian Institute of Technology, Kharagpur-721 302, India Journal of Statistics & Applications