Parity of sets of mutually orthogonal Latin squares

摘要

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 B(k, n) bits. We show that B(k, n) = ((k)(2)) - 1. For the case corresponding to projective planes we prove a tighter bound, namely B(n + 1, n) = (2(k)) when n is odd and B(n + 1, n) = (2(k)) - 1 when n is even. Using the existence of MOLS with subMOLS, we prove that if B(k, n) = ((k)(2)) - 1 then B(k, N) = ((k)(2)) - 1 for all sufficiently large N.