New entanglement-assisted quantum codes from k-Galois dual codes

Liu, Xiusheng; Yu, Lei; Peng, Hu

doi:10.1016/j.ffa.2018.09.001

Cited by 68 publications

(25 citation statements)

References 28 publications

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“…Entanglement-assisted quantum errorcorrecting (EAQEC, for short) codes can boost the rate of transmission by using pre-existing entanglements between the sender and receiver. In [24], Liu, Yu and Hu constructed some new EAQEC codes based on l-Galois dual codes over finite fields. In particular, some new EAQEC codes with better parameters than current EAQEC codes available can be constructed from l-Galois self-orthogonal codes.…”

Section: Introductionmentioning

confidence: 99%

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

Liu

2022

AMC

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<p style='text-indent:20px;'>In this paper, we generalize the notion of self-orthogonal codes to <inline-formula><tex-math id="M1">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonal codes over an arbitrary finite ring. Then, we study the <inline-formula><tex-math id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonality of constacyclic codes of length <inline-formula><tex-math id="M3">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over the finite commutative chain ring <inline-formula><tex-math id="M4">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a prime, <inline-formula><tex-math id="M6">\begin{document}$ u^2 = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> is an arbitrary ring automorphism of <inline-formula><tex-math id="M8">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. We characterize the structure of <inline-formula><tex-math id="M9">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-dual code of a <inline-formula><tex-math id="M10">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic code of length <inline-formula><tex-math id="M11">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M12">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. Further, the necessary and sufficient conditions for a <inline-formula><tex-math id="M13">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic code to be <inline-formula><tex-math id="M14">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-orthogonal are provided. In particular, we determine all <inline-formula><tex-math id="M15">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-self-dual constacyclic codes of length <inline-formula><tex-math id="M16">\begin{document}$ p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M17">\begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>. In the end of this paper, when <inline-formula><tex-math id="M18">\begin{document}$ p $\end{document}</tex-math></inline-formula> is an odd prime, we extend the results to constacyclic codes of length <inline-formula><tex-math id="M19">\begin{document}$ 2 p^s $\end{document}</tex-math></inline-formula>.</p>

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Section: Introductionmentioning

confidence: 99%

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

Liu

2022

AMC

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“…However, the ebit c of EAQEC codes is difficult to calculate. In recent years, a lot of research work has been done for the construction of EAQEC codes and several new families of EAQEC codes have been found by employing different methods [13], [41], [58], [59], [90], [91].…”

Section: Introductionmentioning

confidence: 99%

Quantum MDS and Synchronizable Codes From Cyclic and Negacyclic Codes of Length 2p^s Over F_p^m

2020

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Let p be an odd prime, and F p m is the finite field of p m elements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2p s over F p m are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2p s over finite fields using the Calderbank-Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2p s over F p m. To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes. INDEX TERMS Cyclic codes, repeated-root codes, Hamming distance, MDS codes, quantum MDS codes, quantum synchronizable codes.

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“…Let c = 5 and c = 9, Mustafa and Emre improved the parameters of EAQEC MDS codes with length n further in [28]. Recently, in [24,25], we construct new EAQEC codes by using s-Galois dual codes and parts of them are EAQEC MDS codes.…”

Section: Introductionmentioning

confidence: 99%

A new method for constructing EAQEC MDS codes

Peng¹,

Liu²

2020

Preprint

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Entanglement-assisted quantum error-correcting (EAQEC) codes make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQEC code from any classical linear code, unlike standard quantum error-correcting codes, which can only be constructed from dual-containing codes. However, the number c of pre-shared maximally entangled states is usually calculated by computer search. In this paper, we first give a new formula for calculating the number c of pre-shared maximally entangled states. Then, using this formula, we construct three classes of new entanglement-assisted quantum error-correcting maximum-distance-separable ( EAQEC MDS) codes.

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New entanglement-assisted quantum codes from k-Galois dual codes

Cited by 68 publications

References 28 publications

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

Quantum MDS and Synchronizable Codes From Cyclic and Negacyclic Codes of Length 2p^s Over F_p^m

A new method for constructing EAQEC MDS codes

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New entanglement-assisted quantum codes from k-Galois dual codes

Cited by 68 publications

References 28 publications

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $

Quantum MDS and Synchronizable Codes From Cyclic and Negacyclic Codes of Length 2ps Over F pm

A new method for constructing EAQEC MDS codes

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Quantum MDS and Synchronizable Codes From Cyclic and Negacyclic Codes of Length 2p^s Over F_p^m