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 by Britz, Cavenagh, Mammoliti and Wanless 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'≤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≤v-uh<u/2 for h≥1 then we show that there exists a set of (q^h-1)²/(q-1)-maxMOFS(n;n/q).