Row‐column factorial designs with multiple levels.

An m×nrow‐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. Formally, for any integer q, let [q]={0,1,...,q−1}. The qk (full) factorial design with replication α is the multiset consisting of α occurrences of each element of [q]k; we denote this by α×[q]k. A regularm×nrow‐column factorial design is an arrangement of the elements of α×[q]k into an m×n array (which we say is of typeIk(m,n;q)) such that for each row (column) and fixed vector position i∈[k], each element of [q] occurs n∕q times (respectively, m∕q times). Let m≤n. We show that an array of type Ik(m,n;q) exists if and only if (a) q∣m and q∣n; (b) qk∣mn; (c) (k,q,m,n)≠(2,6,6,6), and (d) if (k,q,m)=(2,2,2) then 4 divides n. Godolphin 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. [ABSTRACT FROM AUTHOR]