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 is an element of {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.