Packings of partial difference sets

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.

arXiv comment41 pages, 2 tables, 1 figure. Minor revisions

References in corpus (4)

• A new family of tight sets in ${\mathcal{Q}}^{+}(5,q)$
• Partial difference sets from quadratic forms and $p$-ary weakly regular bent functions
• Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs, and related geometric substructures
• New constructions of strongly regular Cayley graphs on abelian groups