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 greater than or equal to 7, r-orthogonal Latin squares of order n exist if and only if n less than or equal to r less than or equal to n(2) and r is not an element of {n+1,n(2)-1} with the possible exception of n = 14 and r = n(2)-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 (14(2)-3)-orthogonal Latin squares of order 14, determining the spectrum completely. (C) 2003 Published by Elsevier Science B.V.