Twisted Linearized Reed-Solomon Codes: A Skew Polynomial Framework

Neri, Alessandro

doi:10.48550/arxiv.2105.10451

Cited by 4 publications

(11 citation statements)

References 45 publications

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“…It is clear that they are not linearized Reed-Solomon codes [29], since for q = 2 they are constituted by only one block. Furthermore, twisted linearized Reed-Solomon codes coincide with linearized Reed-Solomon code when q = 2 [31]. Moreover, the constructions presented in [27] have all the same block length, and thus they do not coincide with 2-fold linearized Reed-Solomon codes.…”

Section: Non-homogeneous Case and Multi-linear Setsmentioning

confidence: 96%

“…These codes extend to the sum-rank metric the family of Gabidulin codes in the rank metric and the Reed-Solomon codes in the Hamming metric. They are introduced originally in a vectorial setting while in [31] they have been revisited in a skew polynomial setting.…”

Section: Doubly-extended Linearized Reed-solomon Codesmentioning

confidence: 99%

“…The example of one-weight doubly-extended linearized Reed-Solomon codes can be extended as follows. First we recall the following result from [31] for special sets of parameters. To do this, we introduce the following notation.…”

Section: Non-homogeneous Case and Multi-linear Setsmentioning

confidence: 99%

“…We recall a slightly different notion of sum-rank metric code, in which the codewords are vectors with entries from an extension field F q m rather than matrices over F q . The interested reader is referred to [26,28,29,31,34] for a more detailed description of this setting. The F q -rank of a vector v = (v 1 , .…”

Section: Introductionmentioning

confidence: 99%

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The geometry of one-weight codes in the sum-rank metric

Neri¹,

Santonastaso²,

Zullo³

2021

Preprint

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We provide a geometric characterization of k-dimensional Fqm -linear sum-rank metric codes as tuples of Fq-subspaces of F k q m . We then use this characterization to study one-weight codes in the sum-rank metric. This leads us to extend the family of linearized Reed-Solomon codes in order to obtain a doubly-extended version of them. We prove that these codes are still maximum sum-rank distance (MSRD) codes and, when k = 2, they are one-weight, as in the Hamming-metric case. We then focus on constant rank-profile codes in the sum-rank metric, which are a special family of one weight-codes, and derive constraints on their parameters with the aid of an associated Hamming-metric code. Furthermore, we introduce the n-simplex codes in the sum-rank metric, which are obtained as the orbit of a Singer subgroup of GL(k, q m ). They turn out to be constant rank-profile -and hence one-weight -and generalize the simplex codes in both the Hamming and the rank metric. Finally, we focus on 2-dimensional one-weight codes, deriving constraints on the parameters of those which are also MSRD, and we find a new construction of one-weight MSRD codes when q = 2.

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Section: Non-homogeneous Case and Multi-linear Setsmentioning

confidence: 96%

Section: Doubly-extended Linearized Reed-solomon Codesmentioning

confidence: 99%

Section: Non-homogeneous Case and Multi-linear Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The geometry of one-weight codes in the sum-rank metric

Neri¹,

Santonastaso²,

Zullo³

2021

Preprint

Self Cite

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“…. , N ) was already shown in [21, Theorem 2] (see also [26,Proposition 4.25] for the E-linear sum-rank isometries). Here we deal with the more general case, even though the strategy of the proof is essentially the same.…”

Section: 3mentioning

confidence: 68%

Sum-rank product codes and bounds on the minimum distance

Alfarano¹,

Lobillo²,

Neri³

et al. 2021

Preprint

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The tensor product of one code endowed with the Hamming metric and one endowed with the rank metric is analyzed. This gives a code which naturally inherits the sumrank metric. Specializing to the product of a cyclic code and a skew-cyclic code, the resulting code turns out to belong to the recently introduced family of cyclic-skew-cyclic. A group theoretical description of these codes is given, after investigating the semilinear isometries in the sum-rank metric. Finally, a generalization of the Roos and the Hartmann-Tzeng bounds for the sum rank-metric is established, as well as a new lower bound on the minimum distance of one of the two codes constituting the product code.

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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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Twisted Linearized Reed-Solomon Codes: A Skew Polynomial Framework

Cited by 4 publications

References 45 publications

The geometry of one-weight codes in the sum-rank metric

The geometry of one-weight codes in the sum-rank metric

Sum-rank product codes and bounds on the minimum distance

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

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