A new representation of mutually orthogonal frequency squares

Abstract

We introduce a new representation of MOFS of type $F(m\lambda ;\lambda )$, as a linear combination of $\{0,1\}$ arrays. We use this representation to give an elementary proof of the classical upper bound, together with a structural constraint on a set of MOFS achieving the upper bound. We then use this representation to establish a maximality criterion for a set of MOFS of type $F(m\lambda ;\lambda )$ when $m$ is even and $\lambda$ is odd, which simplifies and extends a previous analysis \cite{ref3} of the case when $m=2$ and $\lambda$ is odd.