Existence of self-orthogonal diagonal Latin squares with a missing subsquare

Abstract

We shall refer to a diagonal Latin square which is orthogonal to its transpose as a self-orthogonal diagonal Latin square, briefly SODLS. This article investigates the spectra of SODLS and SODLS with a missing subsquare. It is found that for any positive integer v except 2, 3 or 6, there exists a SODLS of order v, and for any positive integers v and n, a SODLS of order v missing a subsquare of order n exists if and only if v⩾3 n+2 and v− n is even, except ( v, n)=(8,2) and except possibly for (v,n)∈{(3n+2,n) : n=6,8,10} .