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 , j ∈ N = { 0 , 1 , … , n − 1 } . In this note, we shall prove that an SSSODLS ( n ) exists if and only if n ≡ 0 , 1 , 3 ( mod 4 ) and n ≠ 3 .