A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. It is proved in this paper that there are three pairwise orthogonal diagonal Latin squares of order \(n\) for all \(n \geq 7\) with \(28\) possible exceptions, in which \(118\) is the greatest one.
Citation
L. Zhu. Three Pairwise Orthogonal Diagonal Latin Squares[J], Journal of Combinatorial Mathematics and Combinatorial Computing, Volume 005. 27-40. .