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

Guo, Guanmin; Li, Ruihu; Liu, Yang; Song, Hao

doi:10.48550/arxiv.2202.11457

Cited by 3 publications

(3 citation statements)

References 29 publications

(46 reference statements)

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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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Section: Introductionmentioning

confidence: 99%

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

Chen¹

2022

Preprint

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“…Recently, the authors gave the parity check matrix for the TGRS code and obtained some self-dual TGRS codes with small Singleton defect [14], [27]. More relative results about self-orthogonal TGRS codes can be seen in [28], [29], [11].…”

Section: Introductionmentioning

confidence: 99%

The non-GRS properties for the twisted generalized Reed-Solomon code and its extended code

Zhu¹,

Liao²

2022

Preprint

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In 2017, Beelen et al. firstly introduced twisted generalized Reed-Solomon (in short, TGRS) codes, and constructed a large subclass of MDS TGRS codes. Later, they proved that TGRS code is non-GRS when the code rate is less than one half. In this letter, basing on the dual code of the TGRS code or the extended TGRS code, by using the Schur product, we prove that almost all of TGRS codes and extended TGRS codes are non-GRS when the code rate more than one half.

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“…Furthermore, they presented a sufficient and necessary condition for the (+)-TGRS code to be self-dual, and then constructed several classes of self-dual MDS or NMDS codes [15]. More relative results about self-orthogonal MDS or NMDS TGRS codes can be seen in [8,36,37].…”

Section: Introductionmentioning

confidence: 99%

The $(+)$-extended twisted generalized Reed-Solomon code

Zhu¹,

Liao²

2022

Preprint

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In this paper, we give a parity check matrix for the (+)-extended twisted generalized Reed Solomon (in short, ETGRS) code, and then not only prove that it is MDS or NMDS, but also determine the weight distribution. Especially, based on Schur method, we show that the (+)-ETGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the (+)-ETGRS code to be self-orthogonal, and then construct several classes of self-dual (+)-TGRS codes and almost self-dual (+)-ETGRS codes.Keywords. (+)-extended twisted generalized Reed Solomon codes; MDS codes; NMDS codes; Self-dual codes; Almost self-dual codes.

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

Cited by 3 publications

References 29 publications

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

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

The non-GRS properties for the twisted generalized Reed-Solomon code and its extended code

The $(+)$-extended twisted generalized Reed-Solomon code

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