Row-column factorial designs with strength at least 2

Abstract

The qk (full) factorial design with replication λ is the multi-set consisting of λ occurrences of each element of each q-ary vector of length k; we denote this by λ×[q]k. An m×nrow-column factorial designqk of strength t is an arrangement of the elements of λ×[q]k into an m×n array (which we say is of type Ik(m,n,q,t)) such that for each row (column), the set of vectors therein are the rows of an orthogonal array of degree k, size n (respectively, m), q levels and strength t. Such arrays are used in experimental design. In this context, for a row-column factorial design of strength t, all subsets of interactions of size at most t can be estimated without confounding by the row and column blocking factors.In this manuscript we study row-column factorial designs with strength t≥2. Our results for strength t=2 are as follows. For any prime power q and assuming 2≤M≤N, we show that there exists an array of type Ik(qM,qN,q,2) if and only if k≤M+N, k≤(qM−1)/(q−1) and (k,M,q)≠(3,2,2). We find necessary and sufficient conditions for the existence of Ik(4m,n,2,2) for small parameters. We also show that Ik+α(2αb,2k,2,2) exists whenever α≥2 and 2α+α+1≤k<2αb−α, assuming there exists a Hadamard matrix of order 4b.For t=3 we focus on the binary case. Assuming M≤N, there exists an array of type Ik(2M,2N,2,3) if and only if M≥5, k≤M+N and k≤2M−1. Most of our constructions use linear algebra, often in application to existing orthogonal arrays and Hadamard matrices.