Existence of r-self-orthogonal Latin squares

Two Latin squares of order vv 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)(v). It has been proved that for any integer v⩾28v⩾28, there exists an r  -SOLS(v)(v) if and only if v⩽r⩽v2v⩽r⩽v2 and r∉{v+1,v2-1}r∉{v+1,v2-1}. In this paper, we give an almost complete solution for the existence of r-self-orthogonal Latin squares.