Compositions of double diagonal and cross Latin squares

Abstract

The existence of double diagonal and cross Latin squares for all order (except 2 and 3 in the first case) was shown by Hilton in 1971. Whereas a greatly simplified construction for double diagonal Latin squares was presented immediately afterwards by Gergely in 1972, it has apparently remained open to give simple methods for obtaining larger double diagonal or cross Latin squares for smaller ones. We show that for both types of Latin squares a Kronecker product construction can be devised, using an arbitrary (double-diagonal or cross) Latin square of order pq for any q>= 1. The construction is shown to require only linear time in the size of the constructed object in both cases. We also give a simple direct construction of cross Latin squares of all orders.