Constant Rank-Distance Sets of Hermitian Matrices and Partial Spreads in Hermitian Polar Spaces

Gow, Rod; Lavrauw, Michel; Sheekey, John; Vanhove, Frédéric

doi:10.37236/3534

Cited by 6 publications

(11 citation statements)

References 34 publications

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“…A good overview of MRD-codes can be found in [36]. Similar problems for symmetric, alternating and hermitian matrices have been studied in for example [16], [38], [20], [18].…”

Section: Rank Metric Codesmentioning

confidence: 99%

A new family of linear maximum rank distance codes

Sheekey¹

2016

AMC

Self Cite

200

337

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In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes. Delsarte's duality theoremDefine the symmetric bilinear form b on M m,n (F) by b(X, Y ) := tr(Tr(XY T )),where Tr denotes the matrix trace, and tr denotes the absolute trace from F q to F p , where p is prime and q = p e . Define the Delsarte dual C ⊥ of an F p -linear code C by C ⊥ := {Y : Y ∈ M m,n (F q ), b(X, Y ) = 0 ∀X ∈ C}.

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“…A good overview of MRD-codes can be found in [36]. Similar problems for symmetric, alternating and hermitian matrices have been studied in for example [16], [38], [20], [18].…”

Section: Rank Metric Codesmentioning

confidence: 99%

A new family of linear maximum rank distance codes

Sheekey¹

2016

AMC

Self Cite

200

337

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“…We close this section by showing that the bound for additive codes in Theorem 1 can be surpassed by non-additive codes whenever n is even and d = n. This follows already from [11,Theorem 9]. Here we give a more direct construction.…”

Section: Constructionsmentioning

confidence: 72%

“…It should be noted that related, but different, rank properties of sets of Hermitian matrices have been studied in [9] and [11].…”

Section: Introductionmentioning

confidence: 99%

Hermitian rank distance codes

Schmidt

2017

Des. Codes Cryptogr.

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Abstract. Let X = X(n, q) be the set of n × n Hermitian matrices over F q 2 . It is well known that X gives rise to a metric translation association scheme whose classes are induced by the rank metric. We study d-codes in this scheme, namely subsets Y of X with the property that, for all distinct A, B ∈ Y , the rank of A − B is at least d. We prove bounds on the size of a d-code and show that, under certain conditions, the inner distribution of a d-code is determined by its parameters. Except if n and d are both even and 4 ≤ d ≤ n−2, constructions of d-codes are given, which are optimal among the d-codes that are subgroups of (X, +). This work complements results previously obtained for several other types of matrices over finite fields.

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“…Remark 1. Note that an n-dimensional dual hyperoval is a special case of a constant-distance, constant-dimension subspace code [8], [9], or equivalently, a clique in the graph Γ n−1 , where Γ i is the graph whose vertices are the vertices of Γ, and whose edges are between vertices at distance i in Γ. Note however that not every clique of the correct size in Γ n−1 gives rise to an n-dimensional dual hyperoval.…”

Section: Proof Of Theoremmentioning

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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Constant Rank-Distance Sets of Hermitian Matrices and Partial Spreads in Hermitian Polar Spaces

Cited by 6 publications

References 34 publications

A new family of linear maximum rank distance codes

A new family of linear maximum rank distance codes

Hermitian rank distance codes

Dimensional dual hyperovals in classical polar spaces

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