Row-column factorial designs with strength at least $2$

Abstract

The $q^k$ (full) factorial design with replication $\lambda$ is the multi-set consisting of $\lambda$ occurrences of each element of each $q$-ary vector of length $k$; we denote this by $\lambda \times [q{]}^k$. An $m\times n$ row-column factorial design $q^k$ of strength $t$ is an arrangement of the elements of $\lambda \times [q{]}^k$ into an $m\times n$ array (which we say is of type $I_k(m,n,q,t)$) 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 $t\ge 2$. Our results for strength $t=2$ are as follows. For any prime power $q$ and assuming $2\le M\le N$, we show that there exists an array of type $I_k(q^M,q^N,q,2)$ if and only if $k\le M+N$, $k\le (q^M-1)/(q-1)$ and $(k,M,q)\ne (3,2,2)$. We find necessary and sufficient conditions for the existence of $I_k(4m,n,2,2)$ for small parameters. We also show that $I_{k+\alpha }(2^{\alpha }b,2^k,2,2)$ exists whenever $\alpha \ge 2$ and $2^{\alpha }+\alpha +1\le k<2^{\alpha }b-\alpha$, assuming there exists a Hadamard matrix of order $4b$. For $t=3$ we focus on the binary case. Assuming $M\le N$, there exists an array of type $I_k(2^M,2^N,2,3)$ if and only if $M\ge 5$, $k\le M+N$ and $k\le 2^{M-1}$. Most of our constructions use linear algebra, often in application to existing orthogonal arrays and Hadamard matrices.