On the spectrum of r-self-orthogonal Latin squares

Two Latin squares of order n are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the transpose of the 3rst one, we say that the 3rst square is r-self-orthogonal, denoted by r-SOLS(n). It has been proved that the necessary condition for the existence of an r-SOLS(n) is n6 r6 n and r ∈ {n + 1; n − 1}. Zhu and Zhang conjectured that there is an integer n0 such that for any n¿ n0, there exists an r-SOLS(n) for any r ∈ [n; n2]− {n+ 1; n − 1}. In this paper, we show that n0 6 28. c © 2003 Elsevier B.V. All rights reserved.