Equiorthogonal frequency hypercubes: Preliminary theory (Q1378883)

A frequency square of order \(n\) on \(m\) symbols is an \(n \times n\) array where each symbol appears \(n/m\) times in each row and column. More generally, a Youden frequency hypercube \(\text{YFHC}(m;n^d)\), is a \(d\)-dimensional array such that each \((d-1)\)-subarray contains each of the \(m\) symbols exactly \(n^{d-1}/m\) times. Two \(\text{YFHC}(m;n^d)\)'s are orthogonal if their superposition contains every ordered pair of symbols the same number of times. In the case \(m=n\) and \(d=2\), this reduces to the usual concept of orthogonal latin squares. When \(d \geq 3\), not all subarrays of a frequency object are themselves frequency objects. In the latin case (\(m=n\)) Höhler required each 1-dimensional subarray to have each symbol the same number of times (namely, once). He also strengthened the definition of orthogonality to require corresponding subarrays to be isomorphic or orthogonal. In this paper the author extends Höhler's definitions to frequency arrays. The corresponding stronger form of orthogonality is called equiorthogonality. The author shows that the maximum possible number of mutually equiorthogonal frequency hypercubes (MEFH) of order \(n\) and dimension \(d\) based on \(m\) symbols is \((n-1)^d / (m-1)\). The stronger condition also means that any complete set of MEFH's of dimension \(d\) contains complete sets of MEFH's for all lower dimensions.