Linking systems in nonelementary abelian groups

Linked systems of symmetric designs are equivalent to 3-class Q-antipodal association schemes. Only one infinite family of examples is known, and this family has interesting origins and is connected to important applications. In this paper, we define linking systems, collections of difference sets that correspond to systems of linked designs, and we construct linking systems in a variety of nonelementary abelian groups using Galois rings, partial difference sets, and a product construction. We include some partial results in the final section.