Parity of Sets of Mutually Orthogonal Latin Squares

Every Latin square has three attributes that can be even or odd, but any two of these attributes determines the third. Hence the parity of a Latin square has an information content of 2 bits. We extend the definition of parity from Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the corresponding orthogonal arrays (OA). Suppose the parity of an OA(k,n) has an information content of dim(k,n) bits. We show that dim(k,n) ≤ (k choose 2)-1. For the case corresponding to projective planes we prove a tighter bound, namely dim(n+1,n) ≤ (n choose 2) when n is odd and dim(n+1,n) ≤ (n choose 2)-1 when n is even. Using the existence of MOLS with subMOLS, we prove that if dim(k,n)=(k choose 2)-1 then dim(k,N) = (k choose 2)-1 for all sufficiently large N.

Let the ensemble of an OA be the set of Latin squares derived by interpreting any three columns of the OA as a Latin square. We demonstrate many restrictions on the number of Latin squares of each parity that the ensemble of an OA(k,n) can contain. These restrictions depend on n mod 4 and give some insight as to why it is harder to build projective planes of order n = 2 mod 4 than for n ≠ 2 mod 4. For example, we prove that when n = 2 mod 4 it is impossible to build an OA(n+1,n) for which all Latin squares in the ensemble are isotopic (equivalent to each other up to permutation of the rows, columns and symbols).