MDS or NMDS self-dual codes from twisted generalized Reed–Solomon codes

Huang, Daitao; Yue, Qin; Niu, Yongfeng; Li, Xia

doi:10.1007/s10623-021-00910-7

Cited by 34 publications

(17 citation statements)

References 29 publications

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“…Next according to the check matrix of GT RS k,n [α, v, 1, k−1, η] in [26], we present the following lemma.…”

Section: Hermitian Self-dual (+)-Gtrs Codesmentioning

confidence: 99%

“…GTRS) codes. For other recent studies on GTRS codes, please refer to [26,27,28]. In general, TRS codes are not MDS, nevertheless certain subclasses may be MDS or NMDS which are constructed by a suitable choice of the evaluation points and twist coefficients.…”

Section: Introductionmentioning

confidence: 99%

“…What's more famous is that (+)-twisted Reed-Solomon codes [23], which is called (+)-TRS codes for simplicity. In [26], Huang et al represent the form of check matrix of (+)-GTRS codes.…”

Section: Introductionmentioning

confidence: 99%

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Duality of generalized twisted Reed-Solomon codes and Hermitian self-dual MDS or NMDS codes

Guo¹,

Li²,

Liu³

et al. 2022

Preprint

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Self-dual MDS and NMDS codes over finite fields are linear codes with significant combinatorial and cryptographic applications. In this paper, firstly, we investigate the duality properties of generalized twisted Reed-Solomon (abbreviated GTRS) codes in some special cases. In what follows, a new systematic approach is proposed to draw Hermitian self-dual (+)-GTRS codes. The necessary and sufficient conditions of a Hermitian self-dual (+)-GTRS code are presented. With this method, several classes of Hermitian self-dual MDS and NMDS codes are constructed.Keywords Hermitian self-dual • generalized twisted Reed-Solomon codes • MDS codes • NMDS codes

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“…Next according to the check matrix of GT RS k,n [α, v, 1, k−1, η] in [26], we present the following lemma.…”

Section: Hermitian Self-dual (+)-Gtrs Codesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Duality of generalized twisted Reed-Solomon codes and Hermitian self-dual MDS or NMDS codes

Guo¹,

Li²,

Liu³

et al. 2022

Preprint

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show abstract

“…Remark 1. In [16], the authors showed that the code constructed there is near-MDS if and only if certain subset sum problem has a solution. Here, in our construction, the code is near-MDS if and only if the following subset product problem find a subset S ⊂ D of size k such that η = (−1) k α∈S α has a solution.…”

Section: ) and Its Dualmentioning

confidence: 99%

“…In recent years, constructions of self-dual MDS codes via GRS codes become a hot topic [8,9,19,25,32,33]. After the twisted GRS (TGRS) codes were introduced in [1], the properties of TGRS codes and constructions of self-dual TGRS codes are studied extensively [3,15,16,24,26,34].…”

Section: Introductionmentioning

confidence: 99%

A class of twisted generalized Reed-Solomon codes

Zhang¹,

Zhou²,

Tang³

2022

Preprint

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On Deciding Deep Holes of Reed-Solomon Codes

Cheng

Murray

Lecture Notes in Computer Science

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For generalized Reed-Solomon codes, it has been proved [6] that the problem of determining if a received word is a deep hole is co-NP-complete. The reduction relies on the fact that the evaluation set of the code can be exponential in the length of the code -a property that practical codes do not usually possess. In this paper, we first presented a much simpler proof of the same result. We then consider the problem for standard Reed-Solomon codes, i.e. the evaluation set consists of all the nonzero elements in the field. We reduce the problem of identifying deep holes to deciding whether an absolutely irreducible hypersurface over a finite field contains a rational point whose coordinates are pairwise distinct and nonzero. By applying Schmidt and Cafure-Matera estimation of rational points on algebraic varieties, we prove that the received vector (f (α)) α∈Fq for Reed-Solomon [q, k] q , k < q 1/7−ǫ , cannot be a deep hole, whenever f (x) is a polynomial of degree k + d for 1 ≤ d < q 3/13−ǫ .

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MDS or NMDS self-dual codes from twisted generalized Reed–Solomon codes

Cited by 34 publications

References 29 publications

Duality of generalized twisted Reed-Solomon codes and Hermitian self-dual MDS or NMDS codes

Duality of generalized twisted Reed-Solomon codes and Hermitian self-dual MDS or NMDS codes

A class of twisted generalized Reed-Solomon codes

On Deciding Deep Holes of Reed-Solomon Codes

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