  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.M. Wanless, Mutually orthogonal
binary frequency squares, Electron. J. Combin., 27(#P3.7), 2020, 26 pages] of
the case when $m=2$ and $\lambda$ is odd.
