ORTHOGONAL ARRAYS OF STRENGTH 3 AND SMALL RUN SIZES ANDRIES E. BROUWER, ARJEH M. COHEN, MAN V.M. NGUYEN Abstract. All mixed (or asymmetric) orthogonal arrays of strength 3 with run size at most 64 are determined. 1. Introduction In this paper we study mixed orthogonal arrays of strength 3. Let s1, s2, . , sk be a list of natural numbers, and for each i, let Qsi be a set of size si. For natural numbers t, N, a multiset F of size N whose elements are from Qs1 ×Qs2 ×· · ·×Qsk is called an orthogonal array of strength t, notation OA(N, s1, s2, ··· , sk, t), if t ≤ k, Q and, for every index set I ⊆ {1, ··· , k} of size at most t, each row of i∈I Qsi occurs equally often in the projection of F onto the coordinates indexed by I. We refer to the elements of F as runs, so N is the number of runs of F, also called its run size. The coordinates of Qs1 × Qs2 × · · · × Qsk are called factors, so k is the number of factors. Moreover, si is called the level of the factor i. Instead a b c of s1, s2, s3,... we also write 2 · 3 · 4 ··· , where the exponents a, b, c, . indicate the number of factors at level 2, 3, 4, etc. An orthogonal array is called trivial if it contains each element of Qs1 × Qs2 × · · · × Qsk the same number of times. Orthogonal arrays of strength 2 have been studied extensively. In this paper, we study the case of strength t = 3. We restrict ourselves mainly to N ≤ 64. Theorem 1. For every set of parameters N, s1, s2, . , sk, t with t = 3 and N ≤ 64 such that an orthogonal array OA(N, s1, s2, ··· , sk, t) exists, we construct at least one such array. More precisely, if k = 3 such an array is trivial, and if k > 3 a construction is indicated in Table 1. Of course the existence of OA(N, s1, s2, ··· , sk, t) does not depend on the or- dering of the parameters sj, and we can take them in non-decreasing order if we wish. Sometimes we shall find it useful to consider arrays with si = 1 for some i. An OA(N, 1, s1, s2, ··· , sk, t) is equivalent to an OA(N, s1, s2, ··· , sk, t), provided t ≤ k. We prove the main theorem in three parts. In Section 2, we recall known facts, establish some upper bounds on the parameters of a strength 3 orthogonal array with run size at most 64, and provide further basic material. In Section 3 we de- termine the feasible parameter sets. In Section 4, we provide explicit constructions for all cases mentioned in Table 1. In Section 5, we give the non-existence proof for parameters not appearing in the table. Date: November 1, 2003. Key words and phrases. Fractional factorial designs, orthogonal arrays. 1 2 ANDRIES E. BROUWER, ARJEH M. COHEN, MAN V.M. NGUYEN N Levels Existence Construction Nonexistence 8 2a a ≤ 4 (H) 16 2a · 4 a ≤ 3 (M) 16 2a a ≤ 8 (H) 24 2a · 6 a ≤ 3 (M) 24 2a · 3 a ≤ 4 (M) a = 5 24 2a a ≤ 12 (H) 27 3b b ≤ 4 (L) b = 5 32 2a · 8 a ≤ 3 (M) 32 2a · 42 a ≤ 4 (AD) 32 2a · 4 a ≤ 7 (M) 32 2a a ≤ 16 (H) 36 22 · 32 (T) 40 2a · 10 a ≤ 3 (M) a 40 2 · 5 a ≤ 6 (X1) a = 7 40 2a a ≤ 20 (H) 48 2a · 12 a ≤ 3 (M) 48 2a · 4 · 6 a ≤ 2 (M) a = 3 48 2a · 6 a ≤ 7 (M) a 48 2 · 3 · 4 a ≤ 4 (X2) a = 5 48 2a · 4 a ≤ 11 (M) a 48 2 · 3 a ≤ 9 (X3) a = 10 48 2a a ≤ 24 (H) 54 3b · 6 b ≤ 3 (M) b = 4 a b 54 2 · 3 a ≤ 1, b ≤ 5 (X4) (a, b) = (0, 6) 56 2a · 14 a ≤ 3 (M) 56 2a · 7 a ≤ 6 (J) a = 7 56 2a a ≤ 28 (H) 60 22 · 3 · 5 (T) 64 2a · 16 a ≤ 3 (M) 64 2a · 4 · 8 a ≤ 4 (M) 64 2a · 8 a ≤ 7 (M) 64 4c c ≤ 6 (L) 64 2a · 45 a ≤ 2 (S) a = 3 a 4 64 2 · 4 a ≤ 6 (X5) 64 2a · 43 a ≤ 8 (S) a = 9 64 2a · 42 a ≤ 12 (AD) 64 2a · 4 a ≤ 15 (M) 64 2a a ≤ 32 (H) Table 1. Parameters of orthogonal arrays of strength 3 with N ≤ 64. We briefly explain the table. In the second and third column, a set of levels 2a3b4c ··· is specified for which there exists an orthogonal array of strength 3 and run size N as indicated in the first column. In the fourth column we give a reference to a construction. In the fifth column we give references to nonexistence proofs. Some of the obtained arrays are given in the appendix. We use the terms “factor” and “column”; “fractional design” and “array” alternatively in the whole paper. ORTHOGONAL ARRAYS OF STRENGTH 3 AND SMALL RUN SIZES 3 2. Basic facts We start by fixing some notation and recalling well-known results which will be used throughout the paper. 2.1. Taking the derived design. Proposition 2. Let k ≥ t ≥ 1. From an orthogonal array OA(N, s1, s2, ··· , sk, t), we find, for each choice of column j (1 ≤ j ≤ k) and each choice of symbol s ∈ Qsj , an orthogonal array OA(N/sj, s1, ··· , sj−1, sj+1, ··· , sk, t − 1) by taking the runs that have an s in the j-th position, and deleting that coordinate position. Proof. This is immediately clear from the definition. 2.2. Rao’s generalized bound. We recall the following well-known result (cf. [7], Theorem 9.4, [3], Theorem 2.6.2, [4]). Theorem 3 (Rao’s generalized bounds for mixed orthogonal arrays). Let k ≥ t ≥ 1, and assume an orthogonal array OA(N, s1, s2, ··· , sk, t) exists. • If t is even, then t/2 X X (2.1) N ≥ Πi∈I (si − 1). j=0 |I|=j • If t is odd, then a lower bound for N is found by applying the above bound to the derived design OA(N/sk, s1, s2, ··· , sk−1, t − 1), where sk is the largest among the sj. We will need this result for t = 2 and t = 3. Let us give the explicit form of the above bound in case t = 2. Corollary 4 (Rao’s bound when t = 2). Assume an orthogonal array of the form OA(N, s1, s2, ··· , sk, 2) exists. Then k X (2.2) N ≥ 1 + (si − 1). i=1 For instance, an OA(N, 35.2, 3) has derived design OA(N/3, 34.2, 2), so that N/3 ≥ 1 + 4.(3 − 1) + (2 − 1) = 10, that is, N ≥ 30. These bounds are very well known. It is possible to derive additional information in case equality holds (cf. [3] (5.4.6), [8]). P Proposition 5. Let F be an OA(N, s1, s2, ··· , sk, 2) with N = 1 + (si − 1). P Then for any two runs we have j∈J sj = k − 1, where J is the set of columns where the two runs agree. P Proof. Fix one run, say, the top row. Let the weight of row r be w = j∈J(r) sj, where J(r) is the set of columns where row r agrees with the top row. Let there P be nw rows of weight w. Put σ = (si − 1). Double counting (over all rows other than the top row) now yields X nw = N − 1, w X X N wn = ( − 1)s = Nk − σ − k, w s i w i i 4 ANDRIES E. BROUWER, ARJEH M. COHEN, MAN V.M. NGUYEN X 2 X N X N 2 2 w nw = ( − 1)sisj + ( − 1)si = Nk(k − 1) − (σ + k) + N(σ + k) sisj si w i6=j i so that X Nσ (w − c)2n = (N − 1 − σ) w N − 1 for c = k − σ/(N − 1). Since this last sum is nonnegative, N ≥ 1 + σ. In case equality holds, every weight is equal to c, and c = k − 1. For example, if an OA(24, 221.3, 2) exists, then any two runs have 21 weighted agreements. Since all weights are even except that on the last column, any two runs agree in the last column. Contradiction, so no such array exists. 2.3. Divisibility condition for the run size. For any orthogonal array F, the fact that derived designs exist implies a divisibility condition for the run size of F. Lemma 6. Let k ≥ t ≥ 0. In an OA(N, s1, s2, ··· , sk, t), the run size N is divisible Q by the least common multiple (lcm) of all numbers i∈I si where |I| = t. Proof. This says that the t times derived design has an integral run size. 5 1 For example, in an OA(N, 3 .2 , 3), the run size N must be a multiple of N0 = lcm(3 · 3 · 3, 2 · 3 · 3) = 54. 2.4. Strength 3 transversal designs. Orthogonal arrays are equivalent to trans- versal designs. Here is the version appropriate for mixed arrays. Definition 7. A transversal design (TD) of strength t, is a triple (X, G, B), where X is a set of points; G is a partition of X into groups; and B is a collection of subsets of X called blocks such that: (1) t ≤ |G|, (2) every block meets every group in precisely one point, (3) for every choice of t distinct groups, all t-sets meeting each of those groups in precisely one point are contained in the same number of blocks.