A new representation of mutually orthogonal frequency squares

Mutually orthogonal frequency squares (MOFS) of type $F(m\lambda;\lambda)$ generalize the structure of mutually orthogonal Latin squares: rather than each of $m$ symbols appearing exactly once in each row and in each column of each square, the repetition number is $\lambda \ge 1$. A classical upper bound for the number of such MOFS is $\frac{(m\lambda-1)^2}{m-1}$. 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 [T. Britz, N.J. Cavenagh, A. Mammoliti, I. Wanless, Mutually orthogonal binary frequency squares, arXiv:1912.08972] of the case when $m=2$ and $\lambda$ is odd.