Existence of r-self-orthogonal Latin squares

Abstract

Two Latin squares of order $v$ 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$(v)$. It has been proved that for any integer $v⩾28$, there exists an r-SOLS$(v)$ if and only if $v⩽r⩽v^2$ and $r\notin \{v+1,v^2-1\}$. In this paper, we give an almost complete solution for the existence of r-self-orthogonal Latin squares.