Completing the spectrum of r-orthogonal Latin squares

Two Latin squares of order n are r-orthogonal if their superposition produces exactly r distinct pairs. It has been proved by Belyavskaya, Colbourn and the present authors that for all n⩾7, r-orthogonal Latin squares of order n exist if and only if n⩽r⩽n2 and r∉{n+1,n2−1} with the possible exception of n=14 and r=n2−3. In this paper, we first construct a self-orthogonal Latin square of order 14 which contains certain subarrays. Then we use this square to obtain a pair of (142−3)-orthogonal Latin squares of order 14, determining the spectrum completely.