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).