Mutually orthogonal frequency rectangles

Abstract

A frequency rectangle of type FR(m,n;q) is an m×n matrix such that each symbol from a set of size q appears n/q times in each row and m/q times in each column. Two frequency rectangles of the same type are said to be orthogonal if, upon superimposition, each possible ordered pair of symbols appear the same number of times. A set of k frequency rectangles in which every pair is orthogonal is called a set of mutually orthogonal frequency rectangles, denoted by k–MOFR(m,n;q). We show that a k–MOFR(2,2n;2) and an orthogonal array OA(2n,k,2,2) are equivalent. We also show that an OA(mn,k,2,2) implies the existence of a k–MOFR(2m,2n;2). We construct (4a−2)–MOFR(4,2a;2) assuming the existence of a Hadamard matrix of order 4a.A k–MOFR(m,n;q) is said to be t–orthogonal, if each subset of size t, when superimposed, contains each of the qt possible ordered t-tuples of entries exactly mn/qt times. A set of vectors over a finite field Fq is said to be t-independent if each subset of size t is linearly independent. We describe a method to obtain a set of t–orthogonal k–MOFR(qM,qN,q) corresponding to a set of t–independent vectors in (Fq)M+N. We also discuss upper and lower bounds on the sizes of sets of t–independent vectors and give a table of values for binary vectors of length N⩽16.A frequency rectangle of type FR(n,n;q) is called a frequency square and a set of k mutually orthogonal frequency squares is denoted by k–MOFS(n;q) or k–MOFS(n) when there is no ambiguity about the symbol set. For p an odd prime, we show that there exists a set of (p−1) binary MOFS(2p), hence improving the lower bounds in (Britz et al. 2020) for p⩾19.