On Self Orthogonal Diagonal Latin Squares

This paper considers Latin squares of order \(n\) having \(0, 1, \ldots, n-1\) down the main diagonal and in which the back diagonal is a permutation of these symbols (diagonal squares). It is an open question whether or not such a square which is self-orthogonal (i.e., orthogonal to its transpose) exists for order \(10\). We consider two possible constraints on the general concept: self-conjugate squares and strongly symmetric squares. We show that relative to each of these constraints, a corresponding self-orthogonal diagonal Latin square of order \(10\) does not exist. However, it is easy to construct self-orthogonal diagonal Latin squares of orders \(8\) and \(12\) which satisfy each of the constraints respectively.