9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures

Momihara, Koji; Wang, Qi; Xiang, Qing

doi:10.1515/9783110642094-009

“…Recently, projective two-intersection sets have been intensively studied in the context of intriguing sets in polar spaces (see [4,5,8,12,28,29,30,42], for example). See [23] for a recent survey of intriguing sets, and [66,Sect. 4.5] for a detailed account of the connection between these sets and Latin square type and negative Latin square type PDSs.…”

Section: Proposition 22 (Fourier Inversion Formula) Let G Be An Abelian Group and Letmentioning

confidence: 99%

“…In coding theory, a projective two-weight code is a linear code whose codeword weights take one of exactly two distinct values [13]. In finite geometry, 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]. In graph theory, a strongly regular graph has exactly two distinct eigenvalues [10, Chapter 1].…”

Section: Introductionmentioning

confidence: 99%

Packings of partial difference sets

Jedwab¹,

Li²

2021

Combinatorial Theory

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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.

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“…Recently, projective two-intersection sets have been intensively studied in the context of intriguing sets in polar spaces (see [4,5,8,12,28,29,30,42], for example). See [23] for a recent survey of intriguing sets, and [66,Sect. 4.5] for a detailed account of the connection between these sets and Latin square type and negative Latin square type PDSs.…”

Section: Historical Overviewmentioning

confidence: 99%

“…In coding theory, a projective two-weight code is a linear code whose codeword weights take one of exactly two distinct values [13]. In finite geometry, 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]. In graph theory, a strongly regular graph has exactly two distinct eigenvalues [10, Chapter 1].…”

Section: Introductionmentioning

confidence: 99%

Packings of partial difference sets

Jedwab¹,

Li²

2020

Preprint

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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.

show abstract

“…Recently, strongly regular Cayley graphs with Latin square type or negative Latin square type parameters have been well-studied in relation to geometric substructures, called m-ovoids and i-tight sets, in finite polar spaces. See our survey [14] for recent results and about a relationship between strongly regular Cayley graphs and such geometric substructures.…”

Section: Introductionmentioning

confidence: 99%

New constructions of strongly regular Cayley graphs on abelian groups

Momihara¹

2020

Preprint

Self Cite

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Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a special system of building blocks, called a covering extended building set, and proved that there exists a Paley type partial difference set in an abelian group of order 9 i v 4 for any odd positive integer v > 1 and any i = 0, 1. His result covers all orders of nonelementary abelian groups in which Paley type partial difference sets exist. In this paper, we give new constructions of strongly regular Cayley graphs on abelian groups by extending the theory of building blocks. The constructions are large generalizations of Polhill's construction. In particular, we show that for a positive integer m and elementary abelian groups G i , i = 1, 2, . . . , s, of order q 4 i such that 2m | q i + 1, there exists a decomposition of the complete graph on the abelian group G = G 1 ×G 2 ×• • •×G s by strongly regular Cayley graphs with negative Latin square type parameters (u 2 , c(u + 1), −u + c 2 + 3c, c 2 + c), where u = q 2 1 q 2 2 • • • q 2 s and c = (u − 1)/m. Such strongly regular decompositions were previously known only when m = 2 or G is a p-group. Moreover, we find one more new infinite family of decompositions of the complete graphs by Latin square type strongly regular Cayley graphs. Thus, we obtain many strongly regular graphs with new parameters.

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9. Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures

Cited by 8 publications

References 73 publications

Packings of partial difference sets

Packings of partial difference sets

Packings of partial difference sets

New constructions of strongly regular Cayley graphs on abelian groups

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