Cyclotomic skew partial difference sets and partial difference families

TL;DR

This paper introduces new cyclotomic constructions for skew partial difference sets and related families, expanding known examples and linking them to disjoint and external partial difference families.

Contribution

It presents the first cyclotomic constructions for skew PDSs of Paley type and new Latin square type skew PDSs, broadening the scope of known combinatorial objects.

Findings

First constructions for Paley type skew PDSs

New cyclotomic constructions for DPDFs and EPDFs

Expanded the family of skew PDSs beyond bent partition methods

Abstract

Skew partial difference sets (skew PDSs) are recently-introduced combinatorial objects closely related to partial difference sets (PDSs). To date, only one construction approach for non-trivial skew PDSs is known, using bent partitions: this produces examples of Latin square type. In this paper we show that these examples are not an isolated phenomenon; we present new constructions for families of skew PDSs using cyclotomy in finite fields. We provide the first constructions for skew PDSs of Paley type, and new constructions for Latin square type (with different parameters to those from bent partitions). Moreover, we show how skew PDSs relate to disjoint and external partial difference families (DPDFs/EPDFs), and provide new cyclotomic constructions of both standard and relative DPDFs and EPDFs.