Orthogonal latin squares with subsquares

Abstract

Denote by LS( v, n) a pair of orthogonal latin squares of side v with orthogonal subsquares of side n. It is proved by using a generalized singular direct product that for every odd integer n ⩾ 304 or every even integer n ⩾ 304 in some infinite families, an LS( v, n) exists if and only if v ⩾ 3 n. It is also proved that for every integer n ⩾ 304, an LS( v, n) exists if v > 3 n + 6.