Hermitian Self-Dual GRS and Extended GRS Codes

Guo, Guanmin; Li, Ruihu

doi:10.1109/lcomm.2020.3044893

Cited by 11 publications

(6 citation statements)

References 20 publications

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“…Some of them are new comparing with the constructions in [30,48]. For Hermitian self-dual codes, only few small length Hermitian self-dual GRS codes have been constructed in [36,54]. From Theorem 5.1 there are equivalent arbitrary dimension Hermitian hull GRS codes.…”

Section: Arbitrary Dimension Hull Codes From Self-dual or Hermitian S...mentioning

confidence: 99%

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On Hull-Variation Problem of Equivalent Linear Codes

Chen¹

2022

Preprint

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The intersection C C ⊥ (C C ⊥ h ) of a linear code C and its Euclidean dual C ⊥ (Hermitian dual C ⊥ h ) is called the Euclidean (Hermitian) hull of this code. The construction of an entanglement-assisted quantum code from a linear code over F q or F q 2 depends essentially on the Euclidean hull or the Hermitian hull of this code. Therefore it is natural to consider the hull-variation problem when a linear code C is transformed to an equivalent code v •C. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. A general method to construct hull-decreasing or hull-increasing equivalent linear codes is proposed. We prove that for a nonnegative integer h satisfying 0 ≤ h ≤ n − 1, a linear [2n, n] q self-dual code is equivalent to a linear h-dimension hull code. On the opposite direction we prove that a linear LCD code over F 2 s satisfying d ≥ 2 and d ⊥ ≥ 2 is equivalent to a linear one-dimension hull code under a weak condition. Several new families of negacyclic LCD codes and BCH LCD codes over F 3 are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new EAQEC codes including MDS and almost MDS entanglement-assisted quantum codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.

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Section: Arbitrary Dimension Hull Codes From Self-dual or Hermitian S...mentioning

confidence: 99%

“…There have been a long-time interest in coding theory to construct self-dual, Hermitian self-dual MDS codes. There have been numerous papers on this topic, for example, see [34,36,37,39,42,44,54,63,65,73,74]. Obviously equivalent codes have different dual codes as follows.…”

Section: Introductionmentioning

confidence: 99%

On Hull-Variation Problem of Equivalent Linear Codes

Chen¹

2022

Preprint

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“…There have been numerous papers on this topic, see [23,28,32,45,46,48,49]. However there is few constructions of Hermitian self-dual MDS codes, see [26,27,40]. Obviously equivalent codes have different Hermitian dual codes as follows.…”

Section: Introductionmentioning

confidence: 99%

New MDS Entanglement-Assisted Quantum Codes from MDS Hermitian Self-Orthogonal Codes

Chen¹

2022

Preprint

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The intersection C C ⊥H of a linear code C ⊂ F q 2 and its Hermitian dual C ⊥H is called the Hermitian hull of this code. A linear code C ⊂ F q 2 satisfying C ⊂ C ⊥H is called Hermitian self-orthogonal. Many Hermitian self-orthogonal codes were given for the construction of MDS quantum error correction codes (QECCs). In this paper we prove that for a nonnegative integer h satisfying 0 ≤ h ≤ k, a linear Hermitian self-orthogonal [n, k] q 2 code is equivalent to a linear h-dimension Hermitian hull code. Therefore a lot of new MDS entanglement-assisted quantum error correction (EAQEC) codes can be constructed from previous known Hermitian self-orthogonal codes. Actually our method shows that previous constructed quantum MDS codes from Hermitian self-orthogonal codes can be transformed to MDS entanglement-assisted quantum codes with nonzero consumption parameter c directly. We prove that MDS EAQEC [[n, k, d, c]] q codes with nonzero c parameters and d ≤ n+2 2 exist for arbitrary length n satisfying n ≤ q 2 + 1. Moreover any QECC constructed from kdimensional Hermitian self-orthogonal codes can be transformed to k different EAQEC codes.

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“…Consequently, constructing self-dual NMDS codes remains an open problem for a large range of parameters. As far as we know, however, there are few research results on Hermitian self-dual MDS and NMDS codes, for a few results, see [20,21].…”

Section: Introductionmentioning

confidence: 99%

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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Hermitian Self-Dual GRS and Extended GRS Codes

Cited by 11 publications

References 20 publications

On Hull-Variation Problem of Equivalent Linear Codes

On Hull-Variation Problem of Equivalent Linear Codes

New MDS Entanglement-Assisted Quantum Codes from MDS Hermitian Self-Orthogonal Codes

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

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