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.Keywords: Finite abelian group, packing, partial difference set.Mathematics Subject Classifications: 05B10, 20K01
Highlights
A recurring theme across many diverse areas of mathematics is the natural occurrence of structures for which a key aspect can take one of only two values
A projective two-intersection set is a point set in projective space whose intersection with each hyperplane has one of exactly two distinct sizes [13]; and an m-ovoid or a tight set in a polar space is a point set whose intersection with every tangent hyperplane of the polar space has one of exactly two distinct sizes [10, Chapter 2], [66, Section 4.5]
We refer to a collection of t − 1 > 0 disjoint regular negative Latin square type partial difference sets in an abelian group G of order t2c2, for which the (c − 1)(tc + 1) nonidentity avoided elements of G form a regular negative Latin square type partial difference set in G, as a (c, t − 1) NLP-packing; such a collection has maximum possible size for all c 1
Summary
A recurring theme across many diverse areas of mathematics is the natural occurrence of structures for which a key aspect can take one of only two values. We refer to a collection of t − 1 > 0 disjoint regular (tc, c) negative Latin square type partial difference sets in an abelian group G of order t2c2, for which the (c − 1)(tc + 1) nonidentity avoided elements of G form a regular (tc, c − 1) negative Latin square type partial difference set in G, as a (c, t − 1) NLP-packing; such a collection has maximum possible size for all c 1. [32, Thm. 3.1] [36, Thm. 3.1] [51, Lem. 3.1] [57, Thm. 3.1] [59, Thm. 2.2] [80, Prop. 4.1] [81, Lem. 4.3] [82, Thm. 3.1]
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