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 make usage of cyclotomic Schur rings.

中文翻译：

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通过 Schur 环构建相对差异集的链接系统

在本文中，我们研究相对差异集（RDS）及其链接系统。结果表明，RDS 的封闭链接系统总是按组进行评分。基于此结果，我们还定义了共享相同分级组的 RDS 链接系统的产品。此外，我们推广了 RDS 链接系统的 Davis-Polhill-Smith 构造。最后，我们在有限域上的海森堡群中构建了新的 RDS 链接系统，并在指数\(p^2\)的超特殊p群中构建了 RDS 族。新 RDS 及其连接系统的所有构造都使用分圆 Schur 环。