Row-column factorial designs with strength at least 2

The qk(full) factorial design with replication.is the multiset consisting of.occurrences of each element of each q-ary vector of length k; we denote this by. x[q]k. An m xnrowcolumn factorial designqkof strengthtis an arrangement of the elements of.x[q]kinto an mxnarray (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), qlevels 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 tcan 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 = 2are as follows. For any prime power qand 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 <= (q(M)- 1)/(q- 1) and ( k, M, q) not equal (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 I-k+(alpha)(2(alpha)b, 2(k), 2, 2) exists whenever a = 2and 2(a)+ a+ 1 <= k < 2ab - a, assuming there exists a Hadamard matrix of order 4b. For t = 3we focus on the binary case. Assuming M= N, there exists an array of type Ik(2(M), 2(N), 2, 3) if and only if M >= 5, k <= M+ Nand k <= 2(M-1). Most of our constructions use linear algebra, often inin application to existing orthogonal arrays and Hadamard matrices.

Automated summary

This paper discusses the application of factorial designs in experimental design, particularly focusing on the study of row-column factorial designs. The paper presents necessary and sufficient conditions for the existence of different types of designs and provides examples and results for various parameters.