Constructing linked systems of relative difference sets via Schur rings

In the present paper, we study relative difference sets (RDSs) and linked systems of them. It is shown that a closed linked system of RDSs is always graded by a group. Based on this result, we also define a product of RDS linked systems sharing the same grading group. Further, we generalize the Davis-Polhill-Smith construction of a linked system of RDSs. Finally, we construct new linked system of RDSs in a Heisenberg group over a finite field and family of RDSs in an extraspecial $p$-group of exponent $p^2$. All constructions of new RDSs and their linked systems are based essentially on a usage of cyclotomic Schur rings.