25 new -self-orthogonal Latin squares

Two Latin squares of order n are r -orthogonal if their superposition produces exactly r distinct ordered pairs. If one of the two squares is the transpose of the other, we say that the square is r -self-orthogonal, denoted by r - SOLS ( n ) . It has been proved by Xu and Chang that the necessary and sufficient condition for the existence of an r - SOLS ( n ) is n ≤ r ≤ n 2 and r ∉ { n + 1 , n 2 − 1 } with 26 genuine exceptions and 26 possible exceptions. In this paper, we provide 25 new Latin squares to reduce the possible exceptions from 26 to one, i.e.,  ( n , r ) = ( 14 , 14 2 − 3 ) . We also provide an idempotent incomplete self-orthogonal Latin square (ISOLS) of order 26 with a hole of size 8.