Maximal sets of mutually orthogonal frequency squares

TL;DR

This paper investigates the maximal size of mutually orthogonal frequency squares, extending known results, establishing new bounds, and constructing infinite families of such squares with specific properties.

Contribution

It extends previous bounds on the maximal number of mutually orthogonal frequency squares and provides new constructions for infinite families with fixed cardinality.

Findings

For even n, μ(n) > 2 when n/2 is even.

Existence of sets of k-maxMOFS under divisibility conditions on n.

Infinite families of maximal binary MOFS with fixed size.

Abstract

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$(n;n/q)$.