Maximal sets of mutually orthogonal frequency squares and Doehlert-Klee designs

Abstract

A binary frequency square of type $(n;λ_0,λ_1)$ is a $(0,1)$-matrix of order $n$ with $λ_0$ zeros and $λ_1$ ones in each row and in each column. Two such squares are orthogonal if there are exactly $λ_1^2$ cells where both squares contain ones. A set of binary MOFS is a set of binary frequency squares in which each pair is orthogonal. A set of binary MOFS of type $(n;λ_0,λ_1)$ is type maximal if there is no square of the type $(n;λ_0,λ_1)$ that is orthogonal to every square in the set. A Doehlert-Klee design consists of points $V$ and blocks $B$, where every pair of points occurs in precisely $Λ$ blocks and every point occurs in precisely $R$ blocks, where $R^2=Λ|B|$. We show that sets of binary MOFS are equivalent to a particular kind of Doehlert-Klee design. In a distinct application, Doehlert-Klee designs can also be used to construct sets of binary MOFS that are cyclically generated from their first rows. We use these connections to find new constructions for sets of type-maximal binary MOFS.