Existence of strong symmetric self-orthogonal diagonal Latin squares

A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. A Latin square is self-orthogonal if it is orthogonal to its transpose. A diagonal Latin square L of order n is strongly symmetric, denoted by SSSODLS(n), if L(i,j)+L(n-1-i,n-1-j)=n-1 for all i,[email protected]?N={0,1,...,n-1}. In this note, we shall prove that an SSSODLS(n) exists if and only if n=0,1,3(mod4) and n 3.