On the Existence of Certain SOLS with Holes

Abstract

We consider a pair of MOLS (mutually orthogonal Latin squares) having holes, corresponding to missing sub-MOLS, which are disjoint and spanning. If the two squares are mutual transposes, we say that we have SOLS (self-orthogonal Latin squares) with holes. It is shown that a pair of SOLS with $n$ holes of size $h\ge 2$ exist if and only if $n\ge 4$ and it is also shown that a pair of SOLS with $n$ holes of size $2$ and one hole of size $3$ exist for all $n\ge 4,n\ne 13,15$.

As an application, we prove a result concerning intersection numbers of transversal designs with four groups.