Maximal sets of mutually orthogonal frequency squares

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 $(n;λ)$ if it contains $n/λ$ symbols, each of which occurs $λ$ times per row and $λ$ times per column. In the case when $λ=n/2$ we refer to the frequency square as binary. A set of $k$-MOFS$(n;λ)$ is a set of $k$ frequency squares of type $(n;λ)$ such that when any two of the frequency squares are superimposed, each possible ordered pair occurs equally often. A set of $k$-maxMOFS$(n;λ)$ is a set of $k$-MOFS$(n;λ)$ that is not contained in any set of $(k+1)$-MOFS$(n;λ)$. For even $n$, let $μ(n)$ 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 $μ(n)=1$ if $n/2$ is odd and $μ(n)>1$ if $n/2$ is even. Extending this result, we show that if $n/2$ is even, then $μ(n)>2$. 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 $\limsup μ(n)$ 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$(n;n/q)$.

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