Abelian and non-abelian Paley type group schemes

Chen, Yu Qing; Feng, Tao

doi:10.1007/s10623-012-9640-3

“…The construction of partial difference sets is therefore of great interest. We refer to [13,60] for excellent surveys of partial difference sets and equivalent structures, and to [1,2,4,5,8,9,11,12,14,15,17,18,19,20,21,22,24,25,26,27,28,29,30,31,32,37,38,42,43,44,45,46,47,48,49,50,51,52,54,58,59,61,62,63,64,65,67,68,69,…”

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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“…(3) constructions based on cyclotomy. In particular, the construction by Muzychuk [14] and its generalization by Chen-Feng [2] are very powerful; indeed their constructions yield many inequivalent skew Hadamard difference sets but the group is limited to (F n q , +) with n = 3. For large n > 3, Feng-Xiang's skew Hadamard difference sets [8,13] are the only known class containing infinitely many examples inequivalent to the Paley difference sets.…”

Section: Introductionmentioning

confidence: 99%

“…As far as the author knows, no recursive construction was known while "recursive-like" product constructions were known. The construction given by Muzychuk [14] and its generalization by Chen-Feng [2] needs one skew Hadamard difference set in F q and one "vertically balanced" Paley type partial difference set in F 2 q to construct a skew Hadamard difference set in (F 3 q , +). On the other hand, Chen-Feng's construction [3] needs an "Arasu-Dillon-Player" difference set in F * q n /F * q to construct a skew Hadamard difference set in (F q n , +).…”

Section: Introductionmentioning

confidence: 99%

A Recursive Construction for Skew Hadamard Difference Sets

Momihara

¹

2020

Electron. J. Combin.

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A major conjecture on the existence of abelian skew Hadamard difference sets is: if an abelian group $G$ contains a skew Hadamard difference set, then $G$ must be elementary abelian. This conjecture remains open in general. In this paper, we give a recursive construction for skew Hadamard difference sets in abelian (not necessarily elementary abelian) groups. The new construction can be considered as a result on the aforementioned conjecture: if there exists a counterexample to the conjecture, then there exist infinitely many counterexamples to it.

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“…The construction of partial difference sets is therefore of great interest. We refer to [13,60] for excellent surveys of partial difference sets and equivalent structures, and to [1,2,4,5,8,9,11,12,14,15,17,18,19,20,21,22,24,25,26,27,28,29,30,31,32,37,38,42,43,44,45,46,47,48,49,50,51,52,54,58,59,61,62,63,64,65,67,68,69,…”

Section: Introductionmentioning

confidence: 99%

Packings of partial difference sets

Jedwab¹,

Li²

2020

Preprint

View full text Add to dashboard Cite

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

Abelian and non-abelian Paley type group schemes

Cited by 8 publications

References 22 publications

Packings of partial difference sets

Packings of partial difference sets

A Recursive Construction for Skew Hadamard Difference Sets

Packings of partial difference sets

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