Packings of partial difference sets

TL;DR

This paper studies packings of partial difference sets in abelian groups, providing a unified framework, recursive constructions, and new results on maximum packings, with implications for design theory and related fields.

Contribution

It unifies previous results, introduces recursive and product constructions, and analyzes packings of Latin square type partial difference sets in abelian groups.

Findings

Identifies key subgroups revealing structural information.

Develops recursive lifting constructions for packings.

Provides methods to produce maximum size packings.

Abstract

A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over many years, although often only implicitly. We consider packings of certain Latin square type partial difference sets in abelian groups having identical parameters, the size of the collection being either the maximum possible or one smaller. We unify and extend numerous previous results in a common framework, recognizing that a particular subgroup reveals important structural information about the packing. Identifying this subgroup allows us to formulate a recursive lifting construction of packings in abelian groups of increasing exponent, as well as a product construction yielding packings in the direct product of the starting groups. We also study packings of certain negative Latin square type partial difference sets of maximum possible size in abelian groups, all but one of which have identical parameters, and show how to produce such collections using packings of Latin square type partial difference sets.