Incomplete self-orthogonal latin squares \(ISOLS(6m+6,2m)\) exist for all m (Q757379)

A self-orthogonal latin square of order v (SOLS(v)) is a latin square of order v that is orthogonal to its transpose. An SOLS(v) is known to exist for all v except \(v=2,3,6\). It is known that an incomplete self- orthogonal latin square of order v with an empty subarray of order n (an ISOLS(v,n)), can exist only if \(v\geq 3n+1.\) Using recursive constructions the authors show that an \(ISOLS(6m+6,2m)\) exists for all m. As a consequence there exists an ISOLS(v,n) for all v and n satisfying \(v\geq 3n+1\), except for \(v=6\) and \((v,n)=(8,2)\) and perhaps excepting \((v,n)=(6m+2,2m)\), \(m\geq 2\).