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.

New constructions for disjoint partial difference families and external partial difference families

“…We provide explicit constructions answering both of these questions in the affirmative. We observe that the LP-packings introduced in [22] provide other examples of DPDF/EPDFs in a range of finite abelian groups.…”

Internal and external partial difference families and cyclotomy

“…. , H r such that H i ∩ H j = {0} for i = j is called an (n, r) partial congruence partition of degree r in G (see [17]). More generally, for any group G, a collection of subgroups H 1 , .…”

Constructions and restrictions for balanced splittable Hadamard matrices

“…See the survey article [25] for a detailed treatment of partial difference sets, and [19] for some recent constructions.…”

“…The LP-packings and NLP-packings of partial difference sets introduced in [19] provide examples of (δ, t) partial difference set packings. We now apply Theorem 5.14 to (δ, t) partial difference set packings drawn from the literature [19,32,33] in order to produce infinite families of Hadamard matrices admitting a row decomposition so that the balanced splittable property holds simultaneously with respect to every union of the submatrices of the decomposition. We believe this approach to the construction of balanced splittable Hadamard matrices (or equivalent objects) to be entirely new.…”

“…, a t ), calculate α u = i∈Iu a i and ℓ u = i∈Iu |D i |, and apply Theorem 5.14. (i) By[19, Theorem 5.3], there is a (1, 2 m ) LP-packing in Z 2m 2 relative to Z m 2 . Therefore by Lemma 5.17, there is a (2 m , 2 m +1) partial difference set packing D 1 , .…”