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

Constructing linked systems of relative difference sets via Schur rings