Dimensional doubly dual hyperovals and bent functions

Dempwolff, Ulrich

doi:10.2140/iig.2013.13.149

Cited by 7 publications

(7 citation statements)

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“…For any n ≥ 2 and any even 2-power q, DHOs of rank n over F q are known. There are various constructions of DHOs, see [11,13,14,41,42,45,46] for instance.…”

Section: Dimensional Dual Hyperovals Their Kernels and Nucleimentioning

confidence: 99%

On kernels and nuclei of rank metric codes

2017

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For each rank metric code (Formula presented.), we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When (Formula presented.) is (Formula presented.)-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When (Formula presented.) is a finite field (Formula presented.) and (Formula presented.) is a maximum rank distance code with minimum distance (Formula presented.) or (Formula presented.), the kernel of the associated translation structure is proved to be (Formula presented.). Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over (Formula presented.) must be a finite field; its right nucleus also has to be a finite field under the condition (Formula presented.). Let (Formula presented.) be the DHO-set associated with a bilinear dimensional dual hyperoval over (Formula presented.). The set (Formula presented.) gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to (Formula presented.). Also, its middle nucleus must be a finite field containing (Formula presented.). Moreover, we also consider the kernel and the nuclei of (Formula presented.) where k is a Knuth operation

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“…For any n ≥ 2 and any even 2-power q, DHOs of rank n over F q are known. There are various constructions of DHOs, see [11,13,14,41,42,45,46] for instance.…”

Section: Dimensional Dual Hyperovals Their Kernels and Nucleimentioning

confidence: 99%

On kernels and nuclei of rank metric codes

2017

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“…Dempwolff and Kantor [5] gave a geometric construction leading to many inequivalent examples in Q + (2n−1, 2). Dempwolff [3] gave further examples in W (2n−1, 2) which cannot lie in Q + (2n − 1, 2).…”

Section: Namementioning

confidence: 99%

“…Dempwolff further conjectured in [3] that n-dimensional doubly dual hyperovals over F 2 exist only if n is odd. This remains an open problem.…”

Section: Alternating and Symmetric Doubly Dual Hyperovalsmentioning

confidence: 99%

Dimensional dual hyperovals in classical polar spaces

Sheekey

2015

Des. Codes Cryptogr.

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In this paper we show that n-dimensional dual hyperovals cannot exist in all but one classical polar space of rank n if n is even. This resolves a question posed by Yoshiara. Definitions and preliminariesAn n-dimensional dual arc D in a vector space V (N , q) over a finite field F q is a set of n-dimensional subspaces such that (1) each two intersect in exactly a one-dimensional space;(2) no three intersect non-trivially.It is clear that |D| ≤ q n −1 q−1 + 1. For let S be any element of D. Then the other elements of D intersect S in distinct one-dimensional subspaces, of which there are q n −1 q−1 . If D meets this bound, it is called an n-dimensional dual hyperoval. We will sometimes use the shorthand n-DA and n-DHO.For background and a recent survey of known results and applications, we refer to [16]. Note that the definition therein are in terms of projective spaces, but here we use vector space terminology and notation, following [6]. In this paper we will mostly consider the case N = 2n. In [6], this is required in the definition, but we will not impose this restriction here.

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“…Before determining the stabilizer of A in L(M), we shall show a result which can be obtained by iterating use of equation (3).…”

Section: Automorphismsmentioning

confidence: 99%

“…The subspace of U spanned by all members of S is called the ambient space of S. If there exists a subspace Y of codimension n in the ambient space of S which intersects every member of S at the zero space, we say that S splits over Y . Such a subspace is called a complement to S. We say that S is of split type, if there is a complement to S. For a complement Y to S and a member X of S, we can associate a collection L X (S; Y ) of linear maps from X to Y , which we call the DHO-set following [3]. For the details, see Section 3.2.…”

Section: Introductionmentioning

confidence: 99%

An elementary description of the Mathieu dual hyperoval and its splitness

Yoshiara¹

2015

Innov. Incidence Geom.

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An elementary new construction of a 3-dimensional dual hyperoval M over F 4 is given, as well as an explicit analysis of the structure of its automorphism group. This provides a self-contained introduction to the Mathieu simple group M 22 . The basic properties of M as a dimensional dual hyperoval, e.g. splitness, complements, linear systems, quotients and coverings, are derived from this construction.

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Dimensional doubly dual hyperovals and bent functions

Abstract: We show that dimensional doubly dual hyperovals over F 2 define bent functions. We also discuss some known and a few new examples of dimensional doubly dual hyperovals and study the associated bent functions.

Cited by 7 publications

References 12 publications

On kernels and nuclei of rank metric codes

On kernels and nuclei of rank metric codes

Dimensional dual hyperovals in classical polar spaces

An elementary description of the Mathieu dual hyperoval and its splitness

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