Existence of Self-orthogonal Latin Squares ISOLS\((6m+ 2,2m)\)

An incomplete self-orthogonal latin square of order \(v\) with an empty subarray of order \(n\), an ISOLS(\(v, n\)), can exist only if \(v \geq 3n+1\). It is well known that an ISOLS(\(v, n\)) exists whenever \(v \geq 3n+1\) and \((v,n) \neq (6m+2,2m)\). In this paper we show that an ISOLS(\(6m+2, 2m\)) exists for any \(m \geq 24\).