Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

Feng, Tao; He, Zhong; Chen, Yu Qing

doi:10.1002/jcd.21695

Cited by 6 publications

(8 citation statements)

References 34 publications

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“…Many PDSs within the same group were equivalent to each other: among the 48 nonabelian groups, there were 176 inequivalent PDSs. Note that PDSs were already known in some nonabelian groups [5]. Please find the full PDS database and code at 42ABC/Nonabelian-NLST-PDSs-of-order-64-code-data on Github.…”

Section: Search Resultsmentioning

confidence: 99%

“…PDSs 60 [ 2,12,3,14,4,18,25,61,27,62,35,63,43,44,45,58,59, 60 ], [ 5,6,7,20,30,52,15,56,39,40,46,48,49,50,34,54,23,64 ] 67 [ 7,20,3,37,4,17,18,19,8,25,9,27,51,62,53,55,59, 64 ], [ 5,22,2,12,30,52,14,16,39,40,41,57...…”

Section: Groupmentioning

confidence: 99%

“…However, the fact that many of the PDSs can be broken down into 3 reversible difference sets suggests that there may be a construction method for nonabelian NLST PDSs like demonstrated in Davis and Xiang (2004) for some abelian groups [3]. This construction method could relate to the work done by Feng, He, and Chen (2020) in nonabelian groups with PDSs with the same SRG as the NLST PDS in C 2 4 × C 2 2 [5].…”

Section: Reversible Hadamard Difference Set Breakdown Of Pdssmentioning

confidence: 99%

“…However, very little is known about NLST PDSs in nonabelian groups. Recently, Feng, He, and Chen derived nonabelian PDSs from the Davis-Xiang family [5].…”

Section: Introductionmentioning

confidence: 99%

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Negative Latin Square Type Partial Difference Sets in Nonabelian Groups of Order 64

Brady¹

2022

Preprint

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There exist few examples of negative Latin square type partial difference sets (NLST PDSs) in nonabelian groups. We present a list of 176 inequivalent NLST PDSs in 48 nonisomorphic, nonabelian groups of order 64. These NLST PDSs form 8 nonisomorphic strongly regular graphs. These PDSs were constructed using a combination of theoretical techniques and computer search, both of which are described. The search was run exhaustively on 212/267 nonisomorphic groups of order 64.

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Section: Search Resultsmentioning

confidence: 99%

Section: Groupmentioning

confidence: 99%

Section: Reversible Hadamard Difference Set Breakdown Of Pdssmentioning

confidence: 99%

“…However, very little is known about NLST PDSs in nonabelian groups. Recently, Feng, He, and Chen derived nonabelian PDSs from the Davis-Xiang family [5].…”

Section: Introductionmentioning

confidence: 99%

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Negative Latin Square Type Partial Difference Sets in Nonabelian Groups of Order 64

Brady¹

2022

Preprint

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“…On the other hand, comparatively little is known in the case when

G

is nonabelian. There have been constructions in sporadic cases (see, for instance, [14,15]) and some instances of constructions of infinite families (see [9,10,21]). At the same time, there have been relatively few results dealing with the nonabelian case in general.…”

Section: Introductionmentioning

confidence: 99%

Restrictions on parameters of partial difference sets in nonabelian groups

Swartz

Tauscheck

2020

J of Combinatorial Designs

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A partial difference set S in a finite group G satisfying ∉ S 1 and S S = −1 corresponds to an undirected strongly regular Cayley graph G S Cay(,). While the case when G is abelian has been thoroughly studied, there are comparatively few results when G is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups and are especially effective in groups with a nontrivial center. In particular, these results apply to p-groups, and we are able to rule out the existence of partial difference sets in many instances. K E Y W O R D S partial difference set, strongly regular Cayley graph −1. If the set S is a v k λ μ (, , ,)-PDS, then the Cayley graph G S Cay(,) is a v k λ μ (, , ,)-strongly regular graph (SRG) [17, Proposition 1.1], which means that G S Cay(,) has v vertices, G S Cay(,) is regular of degree k, any two adjacent vertices in G S Cay(,) have exactly λ common neighbors, and any two nonadjacent vertices in G S Cay(,) have exactly How to cite this article: Swartz E, Tauscheck G. Restrictions on parameters of partial difference sets in nonabelian groups.

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Genuinely nonabelian partial difference sets

Polhill,

Davis,

Smith

et al. 2024

J of Combinatorial Designs

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Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

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Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

Cited by 6 publications

References 34 publications

Negative Latin Square Type Partial Difference Sets in Nonabelian Groups of Order 64

Negative Latin Square Type Partial Difference Sets in Nonabelian Groups of Order 64

Restrictions on parameters of partial difference sets in nonabelian groups

Genuinely nonabelian partial difference sets

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