MDS Codes With Hulls of Arbitrary Dimensions and Their Quantum Error Correction

Luo, Gaojun; Cao, Xiwang; Chen, Xiaojing

doi:10.1109/tit.2018.2874953

Cited by 112 publications

(59 citation statements)

References 23 publications

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“…Then for any 1 ≤ k ≤ ⌊ n−1+q q+1 ⌋, Remark 3. The lengths of q-ary MDS EAQECCs constructed in [23] are less than or equal to q + 1. From Theorems 9-12, the lengths of our EAQECCs can be larger than q + 1.…”

Section: Applications To Eaqeccsmentioning

confidence: 99%

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

Fang

et al. 2020

IEEE Trans. Inform. Theory

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In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, which we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. In particular, all possible parameters for the q-ary MDS EAQECCs of length n ≤ q are completely determined and several new classes of q-ary MDS EAQECCs of length n > q are also obtained.

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Section: Applications To Eaqeccsmentioning

confidence: 99%

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

Fang

et al. 2020

IEEE Trans. Inform. Theory

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“…Consequently, they have long been in the focus of coding theorists as well as coding practitioners [3]. There is vigorous research concerning these codes and their decoding algorithms even 60 years after their discovery, which could be documented by the following selected references [6][7][8][9][10][11][12]. Therefore, there are numerous known algorithms for their encoding as well as for decoding.…”

Section: Some Notes On Rs Code Decodingmentioning

confidence: 99%

Erasure decoding of five times extended Reed-Solomon codes

Rakús

Farkas

Palenik

2019

Journal of Electrical Engineering

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“…α , which is different from the one used in [11], [28]. Additionally, by the method, the length of entanglement-assisted quantum codes is more general, so we can obtain more entanglement-assisted quantum MDS codes with minimum distance that is more than q 2 + 1 relative to the ones of [19], [26], [27].…”

Section: Introductionmentioning

confidence: 98%

“…Moreover, in quantum coding theory, how to determine the number of pre-shared maximally entangled states to make the minimum distance of quantum MDS codes larger than q 2 + 1 or even q + 1 is an interesting problem. In [28], although Luo et al studied some classes of entanglement-assisted MDS codes from generalized Reed-Solomon codes under the Euclidean case and the parameters of those codes were new and flexible relative to the ones from [6], [12], [27], [34], the authors just consider the Euclidean construction not Hermitian construction. Very recently, in [11], although Fang et al presented several classes of entanglement-assisted quantum MDS codes by employing the Hermitian hull of generalized Reed-Solomon codes, they did not consider the case of entanglement-assisted quantum MDS codes with length…”

Section: Introductionmentioning

confidence: 99%

Some New Classes of Entanglement-Assisted Quantum MDS Codes Derived From Constacyclic Codes

Chen

Feng

et al. 2019

IEEE Access

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Although quantum maximal-distance-separable (MDS) codes that satisfy the quantum singleton bound have become an important research topic in the quantum coding theory, it is not an easy task to search for quantum MDS codes with the minimum distance that is larger than (q/2) + 1. The pre-shared entanglement between the sender and the receiver can improve the minimum distance of quantum MDS codes such that the minimum distance of some constructed codes achieves (q/2) + 1 or exceeds (q/2) + 1. Meanwhile, how to determine the required number of maximally entangled states to make the minimum distance of quantum MDS codes larger than (q/2) + 1 is an interesting problem in the quantum coding theory. In this paper, we utilize the decomposition of the defining set and q 2-cyclotomic cosets of constacyclic codes with the form q = αm + t or q = αm + α − t and n = (q 2 + 1/α) to construct some new families of entanglement-assisted quantum MDS codes that satisfy the entanglement-assisted quantum singleton bound, where q is an odd prime power and m is a positive integer, while both α and t are positive integers such that α = t 2 + 1. The parameters of these codes constructed in this paper are more general compared with the ones in the literature. Moreover, the minimum distance of some codes in this paper is larger than (q/2) + 1 or q + 1. INDEX TERMS Entanglement-assisted quantum codes, constacyclic codes, maximal-distance-separable (MDS) codes.

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MDS Codes With Hulls of Arbitrary Dimensions and Their Quantum Error Correction

Cited by 112 publications

References 23 publications

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

Erasure decoding of five times extended Reed-Solomon codes

Some New Classes of Entanglement-Assisted Quantum MDS Codes Derived From Constacyclic Codes

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