Row-column factorial designs with multiple levels

An {\em $m\times n$ row-column factorial design} is an arrangement of the elements of a factorial design into a rectangular array. Such an array is used in experimental design, where the rows and columns can act as blocking factors. If for each row/column and vector position, each element has the same regularity, then all main effects can be estimated without confounding by the row and column blocking factors. Formally, for any integer $q$, let $[q]=\{0,1,\dots ,q-1\}$. The $q^k$ (full) factorial design with replication $\alpha$ is the multi-set consisting of $\alpha$ occurrences of each element of $[q]^k$; we denote this by $\alpha\times [q]^k$. A {\em regular $m\times n$ row-column factorial design} is an arrangement of the the elements of $\alpha \times [q]^k$ into an $m\times n$ array (which we say is of {\em type} $I_k(m,n;q)$) such that for each row (column) and fixed vector position $i\in [q]$, each element of $[q]$ occurs $n/q$ times (respectively, $m/q$ times). Let $m\leq n$. We show that an array of type $I_k(m,n;q)$ exists if and only if (a) $q|m$ and $q|n$; (b) $q^k|mn$; (c) $(k,q,m,n)\neq (2,6,6,6)$ and (d) if $(k,q,m)=(2,2,2)$ then $4$ divides $n$. This extends the work of Godolphin (2019), who showed the above is true for the case $q=2$ when $m$ and $n$ are powers of $2$. In the case $k=2$, the above implies necessary and sufficient conditions for the existence of a pair of mutually orthogonal frequency rectangles (or $F$-rectangles) whenever each symbol occurs the same number of times in a given row or column.