Some constructions for tt pairwise orthogonal diagonal Latin squares based on difference matrices

A diagonal Latin square is one whose main and back diagonals are both transversals. Difference matrices, quasi-difference matrices and holey quasi-difference matrices are used to construct tt pairwise orthogonal diagonal Latin squares of order vv (tt PODLS(v)(v)) in this paper. For t=3,4,5t=3,4,5, known existence results for tt PODLS(v)(v) are then updated; for t=3t=3, 4, and 5, there remain respectively 1, 11 and 32 unsolved cases, the largest of which are v=10v=10, 54 and 98 respectively.