Relative Difference Sets Partitioned by Cosets

We explore classical (relative) difference sets intersected with the cosets of a subgroup of small index. The intersection sizes are governed by quadratic Diophantine equations. Developing the intersections in the subgroup yields an interesting class of group divisible designs. From this and the Bose-Shrikhande-Parker construction, we obtain some new sets of mutually orthogonal latin squares. We also briefly consider optical orthogonal codes and difference triangle systems.