Parameter inequalities for orthogonal arrays with mixed levels

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1 Parameter inequalities for orthogonal arrays with mixed levels Wiebke S. Diestelkamp Department of Mathematics University of Dayton Dayton, OH Designs, Codes and Cryptography, Vol. 33 (2004), Abstract An important question in the construction of orthogonal arrays is what the minimal size of an array is when all other parameters are fixed. In this paper, we will provide a generalization of an inequality developed by Bierbrauer for symmetric orthogonal arrays. We will utilize his algebraic approach to provide an analogous inequality for orthogonal arrays having mixed levels and show that the bound obtained in this fashion is often sharper than Rao s bounds. We will also provide a new proof of Rao s inequalities for arbitrary orthogonal arrays with mixed levels based on the same method. Key words. Mixed-level orthogonal array, group character, adjacency operator. AMS Subject Classification: Primary 05B15, Secondary 62K15 1 Introduction Among the first lower bounds on the size of an orthogonal array were those developed by Rao [5] for symmetric orthogonal arrays, and in Rao [6] for orthogonal arrays with mixed levels. Various other bounds have been developed since then; see Hedayat et al. [3] for a survey of the different inequalities that are known thus far. Bierbrauer [2] introduced a lower bound on the size of a symmetric orthogonal array which he found to be better than Rao s for some values of t (the strength of the array), although he provides no specifics about the kind of arrays for which this would be the case. We will provide some examples of parameter combinations for symmetric orthogonal arrays for which his bounds are better than Rao s in Section 2. We will then prove a generalization of Bierbrauer s inequality for mixed-level orthogonal arrays. The proof is based on Bierbrauer s proof in the symmetric case. We will provide some examples for parameter combinations for orthogonal arrays for which this inequality results in better bounds on the size of the array than Rao s inequalities do. In the same paper, Bierbrauer also demonstrated how his method could be used to prove Rao s inequalities for symmetric orthogonal arrays. In Section 3, we provide an analogous 1

2 proof for Rao s inequalities for mixed-level orthogonal arrays. Rao [6] suggests a method of proof in which, as pointed out in Beder [1], only applies to simple arrays (i.e., arrays with no repeated elements). Our proof does not depend on the simplicity of the array and thus holds for arbitrary orthogonal arrays with mixed levels. We will use A to denote the cardinality of a multiset A, taking into account how often each of its element occurs. For example, if A = {a, a, b, c}, then A = 4. Since any set is also a multiset, we will use the same notation for the cardinality of a set. We will denote the multiset consisting of n copies of A by n A. That is, for A as above, 2 A = {a, a, a, a, b, b, c, c}. Finally, Z s will denote the additive cyclic group of integers modulo s. Definition 1.1 Let A 1,..., A k be sets with s 1,..., s k elements, respectively. An orthogonal array of size N, k constraints (or factors), s 1, s 2,..., s k levels and strength t, denoted OA(N, s 1,..., s k, t), is a multiset O on A 1 A k of cardinality N such that the following holds: For any set I = {i 1,..., i t } {1,..., k} there exists a number λ I such that the projection of O onto A i1 A it is the multiset λ I (A i1 A it ). When the s i are not all equal, an orthogonal array of the form OA(N, s 1,..., s k, t) is said to have mixed levels. (Such arrays are also often referred to as asymmetric orthogonal arrays.) If s 1 =... = s k = s, we will denote the array by OA(N, k, s, t) and call it a symmetric orthogonal array. In the proofs provided in this paper, we will make use of the fact that we can view an orthogonal array O of the form OA(N, s 1,..., s k, t) as a function on G = A 1 A k. Namely, for x G, O(x) equals the number of times x occurs in O. Throughout the paper, we will assume that the sets A i are additive groups and utilize the following definition: Definition 1.2 Let x, y A 1 A k. The Hamming weight of x, denoted wt(x), is the number of nonzero components of x. We say that x and y are neighbors, denoted x y, if wt(x y) = 1. 2 A bound for mixed-level arrays Bierbrauer [2] provided the following bound on the size N of a symmetric orthogonal array: Theorem 2.1 Assume that an orthogonal array of the form OA(N, k, s, t) exists. Then N s k ( 1 ) (s 1) k. (1) s (t + 1) Note that this is the dual of the Plotkin bound in the sense that upper bounds on the minimum distance of codes yields lower bounds on the size of orthogonal arrays. In addition to being much easier to compute than Rao s bound, the inequality (1) leads to a larger minimum value for N than Rao s bound for a number of parameter combinations. (Rao s bounds are stated in Section 3, Theorem 3.1.) A few examples are given in Table 1. 2

3 s = 3: t = 3, k = 5 s = 4: t = 4, k = 6 t = 4, k = 6, 7 t = 5, k = 7 t = 5, k = 7, 8 t = 6, k = 8, 9 t = 6, k = 9, 10 t = 7, k = 9, 10 t = 7, k = 9, 10, 11 t = 8, k = 10, 11 t = 8, k = 10, 11, 12 t = 9, k = 11, 12, 13 t = 9, k = 11, 12, 13, 14 t = 10, k = 12, 13, 14 t = 10, k = 12, 13, 14, 15, 16 Table 1: Examples of parameter combinations for OA(N, k, s, t) for which the bound in (2) is sharper than Rao s. Note that it is possible that Rao s bounds and the bound derived from (1) take different values, but still lead to the same minimum value for N, since N also has to be a multiple of s t. All examples listed in Table 1 are instances where the inequality in (1) leads to a greater minimum value for N than Rao s bounds. Early computational results suggest that (1) yields a minimum value for N that is at least as good as Rao s whenever s > 2 and t < k < t(s+1) s 1. Bierbrauer [2] outlined an algebraic proof in which he regards the orthogonal array as a multiset on the group G = (Z s ) k. We will adapt his method to provide the following generalization of Theorem 2.1 for orthogonal arrays with mixed levels. Theorem 2.2 Assume that an orthogonal array of the form OA(N, s 1,..., s k, t) exists. Let s T = s s k, s M = max j (s j ) and s m = min j (s j ). Then ) N (s m ) (1 k s T k. (2) s T + (t k + 1)s M Note that letting s = s 1 = = s k yields the inequality in (1). We have not been able to determine the exact conditions under which the bound given in (2) is superior to Rao s bounds. As is the case for symmetric orthogonal arrays, it is possible that even though the bound in (2) is greater than Rao s bound, both still lead to the same minimum value of N. This is due to the fact that the size of the array must be a multiple of s i1 s i2 s it whenever 0 < i 1 <... < i t k. This immediately leads to the condition N LCM{s i1 s i2 s it : 0 < i 1 < < i t k}. (3) (In fact, N is a multiple of the LCM.) In Table 2, we list just a few examples of orthogonal arrays for which the bound in (2) yields a larger minimum value for N than both Rao s bounds and (3). Proof of Theorem 2.2. Note that if then t + 1 (s M 1) k s M, s T k s T + (t k + 1)s M 1, 3

4 s 1 = 5, s 2 =... = s k = 4: s 1 = 4, s 2 =... = s k = 3: s 1 = 3, s 2 =... = s k = 2: k = 15, t = 12 k = 15, t = 11, 12 k = 19, t = 14 k = 16, t = 12, 13 k = 16, t = 12, 13 k = 20, t = 14, 15, 16 k = 17, t = 13, 14 k = 17, t = 12, 13, 14 k = 21, t = 14, 15, 16 k = 18, t = 14, 15 k = 18, t = 13, 14, 15 k = 22, t = 15, 16, 17, 18 k = 19, t = 15, 16 k = 19, t = 14, 15, 16 k = 23, t = 16, 17, 18 k = 20, t = 16, 17 k = 20, t = 15, 16, 17 k = 24, t = 16, 17, 18, 19, 20 k = 21, t = 16, 17, 18 k = 21, t = 15, 16, 17, 18 k = 25, t = 16, 17, 18, 19, 20 k = 22, t = 17, 18, 19 k = 22, t = 16, 17, 18, 19 k = 23, t = 18, 19, 20 k = 23, t = 17, 18, 19, 20 k = 24, t = 19, 20, 21 k = 24, t = 18, 19, 20, 21 k = 25, t = 20, 21, 22 k = 25, t = 18, 19, 20, 21, 22 Table 2: Examples of parameter combinations for OA(N, k, s 1,..., s k, t) for which the bound in (2) is sharper than both Rao s bound and the condition in (3). and inequality (2) is trivially true. Thus let us assume that t + 1 > (s M 1) k s M. (4) Now, let O be an orthogonal array of the form OA(N, s 1,..., s k, t), and let G = Z s1 Z sk. Then O can be viewed as a multiset on the additive group G. Let ς i be a primitive s i -th root of unity (note that ς i is a complex number). For each z i Z si, the function φ zi (g) = ς gz i i defines a character on the additive group Z si. The characters of G are then the functions φ z (x) = φ zi (x i ) = ς z ix i i, where z = (z 1,..., z k ) G and x = (x 1,..., x k ) G (cf. Ledermann [4]). Consider the space L 2 (G) of complex-valued functions on G with scalar product, given by f, g = 1 f(x)g(x). x G It is a well-known fact that the characters of G form an orthonormal basis of L 2 (G) (see [4]). We will view O as a function on G as described in Section 1. Then we have O (x) = z G µ z φ z (x), where µ z C, and µ z = O, φ z = 1 O(x) x G ς x iz i i. (5) Here the µ z are the Fourier coefficients of O with respect to the orthonormal family {φ z : z G}. 4

5 Now fix z G with wt(z) = t, and let I = {i 1,..., i t } {1,..., k} be the index set corresponding to the nonzero components of z. Let G = Z si1 Z sit, and let π be the projection of G onto G. Denote by O the orthogonal array derived from O by π as follows: for all y G, O (y) = O(x), π(x)=y where the sum is taken over all x O for which π(x) = y. Then µ z = 1 = 1 O(x) y G π(x)=y l I O (y) ς y lz l l. y G l I ς y lz l l Since O has strength t, O consists of all t-tuples (x i1,..., x it ) with i j I j, each occurring λ I times. Thus we have O (y) = λ I for all y G, and so µ z = λ I y G l I ς y lz l l. The function l I ς y lz l l of y = (y i1,..., y it ) is a nontrivial character of G, and thus using the fact that y G l I ς y lz l l = 0, χ(x) = 0 for all nontrivial characters χ of a finite abelian group H. (6) x H Thus µ z = 0 when wt(z) = t. Since O has strength d for every d {1,..., t}, we must have Define the adjacency operator A : L 2 (G) L 2 (G) by µ z = 0 whenever 1 wt(z) t. (7) Af(x) = y x f(y) for f L 2 (G), x G. Then for any z G, Aφ z (x) = k φ z (y) = φ z (y), (8) y x j=1 y B j,x where B j,x = {y G : x and y differ in only the j-th component}. 5

6 Now let y B j,x. If z j = 0, then y j z j = x j z j = 0, and thus y i z i = x i z i i. Therefore and φ z (y) = ς y iz i i = ς x iz i i = φ z (x), y B j,x φ z (y) = B j,x φ z (x) = (s j 1) φ z (x). If z j 0, then φ z (y) = ς y iz i i = ( (ς x iz i i ) ς (y i x i )z i i ) = φ z (x)ς (y j x j )z j j. where Z s j denotes the set Z sj \{0}. But φ z (y) = φ z (x)ς (yj xj)zj j y B j,x y B j,x = φ z (x) ς δzj j δ Z s j = φ z (x) δ Z s j φ zj (δ), δ Z sj φ zj (δ) = 0 by (6), and thus y B j,x φ z (y) = φ z (x). Therefore we have { (sj 1) φ φ z (y) = z (x) if z j = 0, φ y B z (x) if z j 0. j,x Thus (8) becomes Aφ z (x) = (s j 1) wt(z) φ z (x). Letting we have j:z j =0 α z = j:z j =0 (s j 1) wt(z), Aφ z (x) = α z φ z (x) x G. (9) Thus the characters of G are eigenvectors of A, with α z as eigenvalues. Now let s M = max j (s j ) and s T = k s j. Then j=1 α z (s M 1) (k wt(z)) wt(z) = (s M 1)k s M wt(z) (10) 6

7 and α 0 = Next, using (9), it follows that k (s j 1) = s T k. (11) j=1 AO, O = z G α z µ z 2. (12) Recall that if 0 < wt(z) t, then µ z = 0. Therefore AO, O = α 0 µ α z µ z 2. (13) z:wt(z) t+1 Now, by (5), µ 0 = O, φ 0 = 1 x G O(x) = N. Next, by (10), if wt(z) t + 1, then Using (11) and (14), (13) implies that Next, AO, O (s T k) N 2 z G α z (s M 1) k s M (t + 1). (14) 2 + [(s M 1) k s M (t + 1)] µ z 2 = O, O = 1 O (z) 2 N, z G and so (again using the fact that µ z = 0 whenever 0 < wt(z) t) Now, by assumption (4), z:wt(z) t+1 z:wt(z) t+1 µ z 2. (15) µ z 2 N N 2 2. (16) (s M 1) k s M (t + 1) < 0, so multiplying both sides of inequality (16) by (s M 1) k s M (t + 1) and using (15), we see that AO, O (s T k) N 2 ( N 2 + [(s M 1) k s M (t + 1)] N 2 ) 2. (17) But, since O maps into the nonnegative integers, we have AO, O = 1 O (x) O (y) 0. (18) x G y x 7

8 Therefore (17) and (18) imply that 0 (s T k) N [(s M 1) k s M (t + 1)] ( N N 2 ) 2. Let s m = min j (s j ). Solving the last inequality for N and using the fact that (s m ) k, it follows that ) N (s m ) (1 k s T k, s T + (t k + 1)s M as desired. 3 A proof of Rao s inequalities for orthogonal arrays with mixed levels In 1973, Rao [6] stated a generalization of his original inequalities to mixed-level orthogonal arrays: Theorem 3.1 Assume an orthogonal array of the form OA(N, s 1,..., s k, t) exists. If t is even, then t/2 N 1 + (s i 1). (19) If t is odd, then (t 1)/2 N 1 + j=1 I =j i I j=1 I =j i I (s i 1) + max j (s j 1) (s i 1). (20),j / I i I I = t 1 2 The proof of Theorem 3.1 indicated in Rao s paper is valid only for simple orthogonal arrays (i.e. arrays that have no repeated columns) and is carried out in Beder [1]. An alternate proof that does not require the array to be simple is suggested in Hedayat et al. [3]. Bierbrauer [2] provided a proof for Rao s inequalities in the symmetric case that is based on his technique for developing the lower bound in Theorem 2.1. We will extend Bierbrauer s method to prove Rao s inequalities for arrays with mixed levels. This proof also does not require simplicity. Proof of Theorem 3.1. We will use the same definitions as in Theorem 2.2. Let O be an orthogonal array of form OA(N, s 1,..., s k, t). Denote the support of O by P, and consider the space L 2 (P ), with scalar product (f, g) = 1 N O (x) f(x)g (x). x P By restriction of its domain to P, every function in L 2 (G) yields a function in L 2 (P ). We have ( ) dim L 2 (P ) = P O = N. (21) 8

9 For each z G, let us define the function f z as the restriction of the character φ z to P. Then if z, w G, we have (f z, f w ) = 1 O (x) φ z (x)φ w (x) = 1 O (x) φ z w (x) N N x P x P = N O, φ w z = N µ w z. Now, if t is even, let Z 0 = { z G : wt(z) t 2}, and let z, w Z0 be distinct. Then wt(z w) t. Since µ z = 0 whenever 1 wt(z) t, we have (f z, f w ) = N µ z w = 0. Thus for z Z 0, the functions f z are pairwise orthogonal in L 2 (P ) and therefore linearly independent. That means that Z 0 dim ( L 2 (P ) ). Using (21), we have N Z 0. Next, we partition the elements of Z 0 by their weight: Z 0 = enumeration we see that t/2 Z 0 = 1 + (s i 1). j=1 I =j i I This proves inequality (19). If t is odd, fix j {1,..., k}, and let { Z j = z G : either wt(z) < t t + 1, or wt(z) = 2 2 t/2 j=0 {z G : wt (z) = j}. By } and z j 0. Let z, w Z j be distinct. If wt(w) < t 2 or wt(z) < t 2 (or both), then wt(w z) t. If wt(z) =wt(w) = t+1 t 1 2, then z and w have 2 nonzero component besides z j and w j, respectively, and thus wt(w z) 2( t 1 2 ) + 1 = t. In both cases we see that µ z w = 0 by (7). Therefore (f z, f w ) = N µ z w = 0. Thus the functions f z, z Z j, are pairwise orthogonal in L 2 (P ) and hence linearly independent. But that means that Z j dim ( L 2 (P ) ) for all j {1,..., k}. Again using (21), we have N Z j for all j = 1,..., k, and thus N max Z j. (22) j=1,...,k Now, Z j can be partitioned as (t 1)/2 Z j = {z G : wt (z) = i} {z G : wt (z) = j and z j 0}. i=0 9

10 By enumeration we see that (t 1)/2 Z j = 1 + j=1 (s i 1) + (s j 1) I =j i I I = t 1 2,j / I i I (s i 1). (23) Since (23) holds for all j {1,..., k}, this together with (22) proves inequality (20). 4 Concluding remarks The lower bound in (2) appears to be sharper than Rao s bounds for a significant number of parameter combinations. Whether a mixed-level orthogonal array for a given parameter combination and having the minimum size prescribed by the bound actually exists is a question that has yet to be answered. Both Rao s bounds and the linear programming bound for mixed-level orthogonal arrays developed in Sloane and Stufken [7] are much more difficult to compute than the one developed in this paper. A general comparison of the different bounds awaits further investigation. Acknowledgments I would like to thank the referees for a number of suggestions that greatly improved this paper. Thanks also to Jay Beder for his helpful comments. References [1] Jay H. Beder. On Rao s inequalities for arrays of strength d. Utilitas Mathematica, 54:85 109, [2] Jürgen Bierbrauer. Bounds on orthogonal arrays and resilient functions. Journal of Combinatorial Designs, 3: , [3] A. S. Hedayat, N. J. A. Sloane, and John Stufken. Orthogonal arrays: Theory and applications. Springer Verlag, New York, [4] Walter Ledermann. Introduction to group characters. Cambridge University Press, Cambridge, second edition, [5] C. Radhakrishna Rao. Factorial experiments derivable from combinatorial arrangements of arrays. Journal of the Royal Statistical Society, Supplement, IX: , [6] C. Radhakrishna Rao. Some combinatorial problems of arrays and applications to design of experiments. In J. N. Srivastava, editor, A Survey of Combinatorial Theory, chapter 29. North-Holland Publishing Company, [7] N. J. A. Sloane and J. Stufken. A linear programming bound for orthogonal arrays with mixed levels. Journal of Statistical Planning and Inference, 56: ,