Some New Quantum BCH Codes over Finite Fields

Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose–Chaudhuri–Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane’s enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.

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New nonbinary quantum codes with larger distance constructed from BCH codes over 𝔽q2

International Journal of Modern Physics B ◽

10.1142/s0217979217500345 ◽

2017 ◽

Vol 31 (06) ◽

pp. 1750034 ◽

Cited By ~ 1

Author(s):

Gen Xu ◽

Ruihu Li ◽

Qiang Fu ◽

Yuena Ma ◽

Luobin Guo

Keyword(s):

Finite Field ◽

Prime Power ◽

Careful Analysis ◽

Error Correcting Codes ◽

Bch Codes ◽

Quantum Codes ◽

Code Length ◽

Narrow Sense ◽

Cyclotomic Cosets ◽

Quantum Error

This paper concentrates on construction of new nonbinary quantum error-correcting codes (QECCs) from three classes of narrow-sense imprimitive BCH codes over finite field [Formula: see text] ([Formula: see text] is an odd prime power). By a careful analysis on properties of cyclotomic cosets in defining set [Formula: see text] of these BCH codes, the improved maximal designed distance of these narrow-sense imprimitive Hermitian dual-containing BCH codes is determined to be much larger than the result given according to Aly et al. [S. A. Aly, A. Klappenecker and P. K. Sarvepalli, IEEE Trans. Inf. Theory 53, 1183 (2007)] for each different code length. Thus families of new nonbinary QECCs are constructed, and the newly obtained QECCs have larger distance than those in previous literature.

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Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes

International Journal of Quantum Information ◽

10.1142/s0219749914500154 ◽

2014 ◽

Vol 12 (03) ◽

pp. 1450015 ◽

Cited By ~ 22

Author(s):

Liang-Dong Lü ◽

Ruihu Li

Keyword(s):

Linear Codes ◽

Careful Analysis ◽

Galois Field ◽

Error Correcting Codes ◽

Optimal Number ◽

Bch Codes ◽

Design Parameters ◽

Bch Code ◽

Cyclotomic Cosets ◽

Quantum Error

The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.

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Quantum codes over rings

International Journal of Quantum Information ◽

10.1142/s0219749914500208 ◽

2014 ◽

Vol 12 (04) ◽

pp. 1450020 ◽

Cited By ~ 4

Author(s):

Kenza Guenda ◽

T. Aaron Gulliver

Keyword(s):

Linear Codes ◽

Error Correcting Codes ◽

Codes Over Rings ◽

Quantum Codes ◽

Matrix Product ◽

Frobenius Rings ◽

Linear Codes Over Rings ◽

Generalized Gray Map ◽

Css Construction ◽

Quantum Error

This paper considers the construction of quantum error correcting codes from linear codes over finite commutative Frobenius rings. We extend the Calderbank–Shor–Steane (CSS) construction to these rings. Further, quantum codes are extended to matrix product codes. Quantum codes over 𝔽pk are also obtained from linear codes over rings using the generalized Gray map.

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An explicit construction of quantum codes from one-generator generalized quasi-cyclic codes

MATEC Web of Conferences ◽

10.1051/matecconf/202133604001 ◽

2021 ◽

Vol 336 ◽

pp. 04001

Author(s):

Yu Yao ◽

Yuena Ma ◽

Husheng Li ◽

Jingjie Lv

Keyword(s):

Cyclic Codes ◽

Explicit Construction ◽

Error Correcting Codes ◽

Inner Product ◽

Quantum Codes ◽

Dual Codes ◽

Index 2 ◽

Quantum Error ◽

Hermitian Inner Product ◽

Quasi Cyclic Codes

In this paper, we take advantage of a class of one-generator generalized quasi-cyclic (GQC) codes of index 2 to construct quantum error-correcting codes. By studying the form of Hermitian dual codes and their algebraic structure, we propose a sufficient condition for self-orthogonality of GQC codes with Hermitian inner product. By comparison, the quantum codes we constructed have better parameters than known codes.

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ON TWO PROBLEMS OF ASYMMETRIC QUANTUM CODES

International Journal of Modern Physics B ◽

10.1142/s0217979214500179 ◽

2014 ◽

Vol 28 (06) ◽

pp. 1450017 ◽

Cited By ~ 6

Author(s):

RUIHU LI ◽

GEN XU ◽

LUOBIN GUO

Keyword(s):

Cyclic Codes ◽

Error Correcting Codes ◽

Bch Code ◽

Quantum Codes ◽

Quantum Code ◽

Asymmetric Quantum ◽

Asymmetric Quantum Codes ◽

Special Cases ◽

Quantum Error ◽

Better Than

In this paper, we discuss two problems on asymmetric quantum error-correcting codes (AQECCs). The first one is on the construction of a [[12, 1, 5/3]]2 asymmetric quantum code, we show an impure [[12, 1, 5/3 ]]2 exists. The second one is on the construction of AQECCs from binary cyclic codes, we construct many families of new asymmetric quantum codes with dz> δ max +1 from binary primitive cyclic codes of length n = 2m-1, where δ max = 2⌈m/2⌉-1 is the maximal designed distance of dual containing narrow sense BCH code of length n = 2m-1. A number of known codes are special cases of the codes given here. Some of these AQECCs have parameters better than the ones available in the literature.

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New quantum codes from skew constacyclic codes

Advances in Mathematics of Communications ◽

10.3934/amc.2021028 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Ram Krishna Verma ◽

Om Prakash ◽

Ashutosh Singh ◽

Habibul Islam

Keyword(s):

Finite Field ◽

Gray Map ◽

Quantum Codes ◽

Constacyclic Codes ◽

Dual Codes ◽

Chain Ring ◽

Css Construction ◽

Positive Integers

<p style='text-indent:20px;'>For an odd prime <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula> and positive integers <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \ell $\end{document}</tex-math></inline-formula>, let <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_{p^m} $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M5">\begin{document}$ p^{m} $\end{document}</tex-math></inline-formula> elements and <inline-formula><tex-math id="M6">\begin{document}$ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $\end{document}</tex-math></inline-formula>. Thus <inline-formula><tex-math id="M7">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula> is a finite commutative non-chain ring of order <inline-formula><tex-math id="M8">\begin{document}$ p^{2^{\ell} m} $\end{document}</tex-math></inline-formula> with characteristic <inline-formula><tex-math id="M9">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, we aim to construct quantum codes from skew constacyclic codes over <inline-formula><tex-math id="M10">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula>. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.</p>

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Entanglement-assisted quantum codes achieving the quantum singleton bound but violating the quantum hamming bound

Quantum Information and Computation ◽

10.26421/qic14.13-14-4 ◽

2014 ◽

Vol 14 (13&14) ◽

pp. 1107-1116

Author(s):

Ruihu Li ◽

Luobin Guo ◽

Zongben Xu

Keyword(s):

Error Correction ◽

Infinite Family ◽

Error Correcting Codes ◽

Quantum Codes ◽

Singleton Bound ◽

Minimum Distances ◽

Quantum Error

We give an infinite family of degenerate entanglement-assisted quantum error-correcting codes (EAQECCs) which violate the EA-quantum Hamming bound for non-degenerate EAQECCs and achieve the EA-quantum Singleton bound, thereby proving that the EA-quantum Hamming bound does not asymptotically hold for degenerate EAQECCs. Unlike the previously known quantum error-correcting codes that violate the quantum Hamming bound by exploiting maximally entangled pairs of qubits, our codes do not require local unitary operations on the entangled auxiliary qubits during encoding. The degenerate EAQECCs we present are constructed from classical error-correcting codes with poor minimum distances, which implies that, unlike the majority of known EAQECCs with large minimum distances, our EAQECCs take more advantage of degeneracy and rely less on the error correction capabilities of classical codes.

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ON OPTIMAL QUANTUM CODES

International Journal of Quantum Information ◽

10.1142/s0219749904000079 ◽

2004 ◽

Vol 02 (01) ◽

pp. 55-64 ◽

Cited By ~ 117

Author(s):

MARKUS GRASSL ◽

THOMAS BETH ◽

MARTIN RÖTTELER

Keyword(s):

Minimum Distance ◽

Prime Power ◽

Quantum Systems ◽

Error Correcting Codes ◽

Mds Codes ◽

Quantum Codes ◽

Maximum Distance Separable ◽

Shortened Codes ◽

Quantum Error ◽

Quantum Mds Codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q2, q2-2d+2, d〛q for 1≤d≤q which additionally give rise to shortened codes 〚q2-s, q2-2d+2-s, d〛q for some s.

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