Row‐column factorial designs with multiple levels

An m × 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. Formally, for any integer q , let [ q ] = { 0 , 1 , … , q − 1 } . The q k (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 regular m × n row‐column factorial design is an arrangement of the elements of α × [ q ] k into an m × n array (which we say is of typeI k ( 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 I k ( m , n ; q ) exists if and only if (a) q ∣ m and q ∣ n ; (b) q k ∣ m n ; (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.