25 new [formula omitted]-self-orthogonal Latin squares

Abstract

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≤n2 and r∉{n+1,n2−1} 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)=(14,142−3). We also provide an idempotent incomplete self-orthogonal Latin square (ISOLS) of order 26 with a hole of size 8.