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.

Cyclotomic association schemes of broad classes and applications to the construction of combinatorial structures

“…For instance, the difference set in Theorem 3.8 (note that difference sets are special cases of partial difference sets) is a union of products of trivial partial difference sets consisting of a single element or all the elements in F 2 2 . Some works in which product constructions of partial difference sets appear in the mathematical literature are [9,10,15,16,17,18]. For instance, the parameters v, k, λ, µ of the 5 previously shown sporadic partial difference sets corresponding to e = 1, n = 4 can be obtained with a product construction in the following two results, that can be found in [10] and also in [16]: 3 ) negative Latin square type and one is of (n, n 3 − 1) negative Latin square type.…”

“…Some works in which product constructions of partial difference sets appear in the mathematical literature are [9,10,15,16,17,18]. For instance, the parameters v, k, λ, µ of the 5 previously shown sporadic partial difference sets corresponding to e = 1, n = 4 can be obtained with a product construction in the following two results, that can be found in [10] and also in [16]: 3 ) negative Latin square type and one is of (n, n 3 − 1) negative Latin square type.…”