Additive Rank Metric Codes

Otal, Kamil; Özbudak, Ferruh

doi:10.1109/tit.2016.2622277

Cited by 48 publications

(49 citation statements)

References 13 publications

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“…After the submission of this paper, Otal andÖzbudak [19] proved that the twisted Gabidulin codes can be further generalized into additive MRD codes. Moreover, several new families of MRD codes consisting of n × n matrices have been constructed, including…”

Section: Introductionmentioning

confidence: 99%

Generalized twisted Gabidulin codes

Lunardon

Trombetti

Zhou

2018

Journal of Combinatorial Theory, Series A

120

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Let C be a set of m by n matrices over Fq such that the rank of A − B is at least d for all distinct A, B ∈ C. Suppose that m n. If #C = q n(m−d+1) , then C is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary 1 < d < m − 1. One was found by Delsarte (1978) and Gabidulin (1985) independently, and it was later generalized by Kshevetskiy and Gabidulin (2005). We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016), and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family. q . When d(C) = d, it is well-known that #C ≤ q max{m,n}(min{m,n}−d+1) ,

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Section: Introductionmentioning

confidence: 99%

Generalized twisted Gabidulin codes

Lunardon

Trombetti

Zhou

2018

Journal of Combinatorial Theory, Series A

120

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“…This group is called the proper automorphism group of C and denoted by Aut (p) (C). Similarly, the full automorphism group of C is defined as the group generated by the union of Aut (p) (C) and the set of [g(x), h(x)] couples satisfying (12)…”

Section: 2mentioning

confidence: 99%

“…• 2016: A more general family including both generalized Gabidulin codes and twisted Gabidulin codes, known as generalized twisted Gabidulin codes, was remarked in [15] and investigated in [9]. In the literature, there are also non-linear constructions of MRD codes (see for instance [2,5,12,13]). However, in this paper we only focus on linear codes since we are interested in duality questions.…”

mentioning

confidence: 99%

Self-duality of generalized twisted Gabidulin codes

Otal¹,

Özbudak²,

Willems³

2018

Advances in Mathematics of Communications

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Self-duality of Gabidulin codes was investigated in [10] and the authors provided an if and only if condition for a Gabidulin code to be equivalent to a self-dual maximum rank distance (MRD) code. In this paper, we investigate the analog problem for generalized twisted Gabidulin codes (a larger family of linear MRD codes including the family of Gabidulin codes). We observe that the condition presented in [10] still holds for generalized Gabidulin codes (an intermediate family between Gabidulin codes and generalized twisted Gabidulin codes). However, beyond the family of generalized Gabidulin codes we observe that some additional conditions are required depending on the additional parameters. Our tools are similar to those in [10] but we also use linearized polynomials, which leads to further tools and direct proofs.

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“…Their natural analogue in the rank metric is represented by maximum rank distance (MRD) codes, that are defined analogously as codes that attain the Singleton bound with equality. Although it was proven that there are plenty of MRD codes that are linear over the extension field [20,3], only few new families have been discovered recently [28,15,22,9,5].…”

Section: Introductionmentioning

confidence: 99%

Systematic encoders for generalized Gabidulin codes and the q-analogue of Cauchy matrices

Neri

2020

Linear Algebra and its Applications

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We characterize the generator matrix in standard form of generalized Gabidulin codes. The parametrization we get for the non-systematic part of this matrix coincides with the qanalogue of generalized Cauchy matrices, leading to the definition of generalized rank Cauchy matrices. These matrices can be represented very conveniently and their representation allows to define new interesting subfamilies of generalized Gabidulin codes whose generator matrix is a structured matrix. In particular, as an application, we construct Gabidulin codes whose generator matrix is the concatenation of an identity block and a Toeplitz/Hankel matrix. In addition, our results allow to give a new efficient criterion to verify whether a rank metric code of dimension k and length n is a generalized Gabidulin code. This criterion is only based on the computation of the rank of one matrix and on the verification of the linear independence of two sets of elements and it requires O(k 2 n) field operations.

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Additive Rank Metric Codes

Cited by 48 publications

References 13 publications

Generalized twisted Gabidulin codes

Generalized twisted Gabidulin codes

Self-duality of generalized twisted Gabidulin codes

Systematic encoders for generalized Gabidulin codes and the q-analogue of Cauchy matrices

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