Orthogonal trades in complete sets of MOLS

Abstract

Let B-p be the Latin square given by the addition table for the integers modulo an odd prime p (i.e. the Cayley table for (Z(p), +)). Here we consider the properties of Latin trades in B-p which preserve orthogonality with one of the p-1 MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in p for the number of times each symbol occurs in such a trade, with an overall lower bound of (logp)(2) / log log p for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orthomorphism by a small amount. We also show that any transversal in B-p hits the main diagonal either p or at most p - log(2) p - 1 times. Finally, if p equivalent to 1 (mod 6) we show the existence of a Latin square which is orthogonal to B-p and which contains a 2 x 2 subsquare.