  A frequency square is a square matrix in which each row and column is a
permutation of the same multiset of symbols. A frequency square is of type
$(n;\lambda)$ if it contains $n/\lambda$ symbols, each of which occurs
$\lambda$ times per row and $\lambda$ times per column. In the case when
$\lambda=n/2$ we refer to the frequency square as binary. A set of
$k$-MOFS$(n;\lambda)$ is a set of $k$ frequency squares of type $(n;\lambda)$
such that when any two of the frequency squares are superimposed, each possible
ordered pair occurs equally often.
  A set of $k$-maxMOFS$(n;\lambda)$ is a set of $k$-MOFS$(n;\lambda)$ that is
not contained in any set of $(k+1)$-MOFS$(n;\lambda)$. For even $n$, let
$\mu(n)$ be the smallest $k$ such that there exists a set of
$k$-maxMOFS$(n;n/2)$. It was shown in [Electron. J. Combin. 27(3) (2020), P3.7]
that $\mu(n)=1$ if $n/2$ is odd and $\mu(n)>1$ if $n/2$ is even. Extending this
result, we show that if $n/2$ is even, then $\mu(n)>2$. Also, we show that
whenever $n$ is divisible by a particular function of $k$, there does not exist
a set of $k'$-maxMOFS$(n;n/2)$ for any $k'\le k$. In particular, this means
that $\limsup \mu(n)$ is unbounded. Nevertheless we can construct infinite
families of maximal binary MOFS of fixed cardinality. More generally, let
$q=p^u$ be a prime power and let $p^v$ be the highest power of $p$ that divides
$n$. If $0\le v-uh<u/2$ for $h\ge1$ then we show that there exists a set of
$(q^h-1)^2/(q-1)$-maxMOFS$(n;n/q)$.
