On the spectrum of r-self-orthogonal Latin squares Original Research Article

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 first one, we say that the first 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 n⩽r⩽n2 and r∉{n+1,n2−1}. 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,n2−1}. In this paper, we show that n0⩽28.