Mutually orthogonal binary frequency squares

A frequency square is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only binary frequency squares of order n with n/2 zeros and n/2 ones in each row and column. Two such frequency squares are orthogonal if, when superimposed, each of the 4 possible ordered pairs of entries occurs equally often. In this context we say that a set of k-MOFS(n) is a set of k binary frequency squares of order n in which each pair of squares is orthogonal.

A set of k-MOFS(n) must satisfy k≤(n-1)², and any set of MOFS achieving this bound is said to be complete. For any n for which there exists a Hadamard matrix of order n we show that there exists at least 2^{n²/4-O(n log n)} isomorphism classes of complete sets of MOFS(n). For 2<n=2 mod 4 we show that there exists a set of 17-MOFS(n) but no complete set of MOFS(n).

A set of k-maxMOFS(n) is a set of k-MOFS(n) that is not contained in any set of (k+1)-MOFS(n). By computer enumeration, we establish that there exists a set of k-maxMOFS(6) if and only if k in {1,17} or 5 ≤ k ≤ 15. We show that up to isomorphism there is a unique 1-maxMOFS(n) if n=2 mod 4, whereas no 1-maxMOFS(n) exists for n=0 mod 4. We also prove that there exists a set of 5-maxMOFS(n) for each order n=2 mod 4 where n ≥ 6.