Difference set constructions of DRADs and association schemes

Doubly Regular Asymmetric Digraphs (DRAD) with rank 4 automorphism groups were previously thought to be rare. We exhibit difference sets in Galois Rings that can be used to construct an infinite family of DRADs with rank 4 automorphism groups. In addition, we construct difference sets in groups for all r⩾2 that can be used to construct DRADs and nonsymmetric 3-class imprimitive association schemes. Finally, we prove a new product construction for difference sets so that the resulting difference sets can be used to build nonsymmetric 3-class imprimitive association schemes.