Internal and external partial difference families and cyclotomy

TL;DR

This paper introduces new classes of difference families called disjoint and external partial difference families, explores their properties, constructions, and connections to other combinatorial structures, including cyclotomy-based methods.

Contribution

It defines disjoint and external partial difference families, develops a unified cyclotomic framework for their construction, and links them to existing difference structures and PDSs.

Findings

Established properties and constructions of DPDFs and EPDFs.

Connected DPDFs and EPDFs to collections of PDSs.

Developed cyclotomic methods for constructing DPDFs and EPDFs.

Abstract

We introduce the concept of a disjoint partial difference family (DPDF) and an external partial difference family (EPDF), a natural generalisation of the much-studied structures of disjoint difference family (DDF), external difference family (EDF) and partial difference set (PDS). We establish properties and constructions, and indicate connections to other recently-studied combinatorial structures. We show how DPDFs and EPDFs may be formed from collections of PDSs, and also present cyclotomic methods yielding DPDFs and EPDFs whose component sets are not in general PDSs. As part of this, we develop a unified framework encompassing various known constructions for cyclotomic difference structures, which also yields new results on DDFs and EDFs.