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

2017 ◽  
Vol 31 (06) ◽  
pp. 1750034 ◽  
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.

NEW QUANTUM CODES CONSTRUCTED FROM A CLASS OF IMPRIMITIVE BCH CODES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350006 ◽  
Author(s):  
YANG LIU ◽  
YUENA MA ◽  
YOUQIAN FENG ◽  
RUIHU LI
Keyword(s):  
Lower Bound ◽  
Prime Power ◽  
Careful Analysis ◽  
Bch Codes ◽  
Bch Code ◽  
Quantum Codes ◽  
Narrow Sense ◽  
Cyclotomic Cosets ◽  
Steane Construction ◽  
Better Than

By a careful analysis on cyclotomic cosets, the maximal designed distance δnew of narrow-sense imprimitive Euclidean dual containing q-ary BCH code of length [Formula: see text] is determined, where q is a prime power and l is odd. Our maximal designed distance δnew of dual containing narrow-sense BCH codes of length n improves upon the lower bound δmax for maximal designed distances of dual containing narrow-sense BCH codes given by Aly et al. [IEEE Trans. Inf. Theory53 (2007) 1183]. A series of non-narrow-sense dual containing BCH codes of length n, including the ones whose designed distances can achieve or exceed δnew, are given, and their dimensions are computed. Then new quantum BCH codes are constructed from these non-narrow-sense imprimitive BCH codes via Steane construction, and these new quantum codes are better than previous results in the literature.

Hermitian dual containing BCH codes and Construction of new quantum codes

2013 ◽  
Vol 13 (1&2) ◽  
pp. 21-35
Author(s):  
Ruihu Li ◽  
Fei Zou ◽  
Yang Liu ◽  
Zongben Xu
Keyword(s):  
Prime Power ◽  
Careful Analysis ◽  
Bch Codes ◽  
Quantum Codes ◽  
Narrow Sense ◽  
Cyclotomic Cosets

Let $q\geq 3$ be a prime power. Maximal designed distances of primitive Hermitian dual containing $q^{2}$-ary BCH codes (narrow-sense or non-narrow-sense) are determined by a careful analysis of properties of cyclotomic cosets. Non-narrow-sense BCH codes which achieve these maximal designed distances are presented, and a sequence of nested non-narrow-sense BCH codes that contain these BCH codes with maximal designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these non-narrow-sense BCH codes. The nonbinary quantum BCH codes presented here have better parameters than those quantum BCH codes available in the literature.

Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes

2014 ◽  
Vol 12 (03) ◽  
pp. 1450015 ◽  
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.

scholarly journals Some New Quantum BCH Codes over Finite Fields

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 712
Author(s):  
Lijuan Xing ◽  
Zhuo Li
Keyword(s):  
Finite Fields ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Quantum Codes ◽  
Dual Codes ◽  
Cyclotomic Cosets ◽  
New Family ◽  
Consecutive Integers ◽  
Css Construction ◽  
Quantum Error

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.

New optimal asymmetric quantum codes constructed from constacyclic codes

2017 ◽  
Vol 31 (05) ◽  
pp. 1750030 ◽  
Author(s):  
Gen Xu ◽  
Ruihu Li ◽  
Luobin Guo ◽  
Liangdong Lü
Keyword(s):  
Prime Power ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Code Length ◽  
Theor Phys ◽  
Singleton Bound ◽  
Asymmetric Quantum ◽  
Asymmetric Quantum Codes ◽  
Quantum Error

In this paper, we propose the construction of asymmetric quantum codes from two families of constacyclic codes over finite field [Formula: see text] of code length [Formula: see text], where for the first family, [Formula: see text] is an odd prime power with the form [Formula: see text] ([Formula: see text] is integer) or [Formula: see text] ([Formula: see text] is integer) and [Formula: see text]; for the second family, [Formula: see text] is an odd prime power with the form [Formula: see text] or [Formula: see text] ([Formula: see text] is integer) and [Formula: see text]. As a result, families of new asymmetric quantum codes [Formula: see text] with [Formula: see text] distance larger than [Formula: see text] are obtained, which are not covered by the asymmetric quantum error-correcting codes (AQECCs) in Refs. 32 and 33 [J.-Z. Chen, J.-P. Li and J. Lin, Int. J. Theor. Phys. 53, 72 (2014); L. Wang and S. Zhu, Int. J. Quantum Inf. 12, 1450017 (2014)] that [Formula: see text]. Also, all the newly obtained asymmetric quantum codes are optimal according to the singleton bound for asymmetric quantum codes.

scholarly journals ON OPTIMAL QUANTUM CODES

2004 ◽  
Vol 02 (01) ◽  
pp. 55-64 ◽  
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.

New quantum constacyclic codes with length n=2(qm+1)

2019 ◽  
Vol 17 (07) ◽  
pp. 1950057
Author(s):  
Junli Wang ◽  
Ruihu Li ◽  
Yang Liu ◽  
Hao Song
Keyword(s):  
Prime Power ◽  
Bch Codes ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Cyclotomic Cosets ◽  
The Given

By studying the properties of [Formula: see text]-cyclotomic cosets, the maximum designed distances of Hermitian dual-containing constacyclic Bose–Chaudhuri–Hocquenghem (BCH) codes with length [Formula: see text] are determined, where [Formula: see text] is an odd prime power and [Formula: see text] is an integer. Further, their dimensions are calculated precisely for the given designed distance. Consequently, via Hermitian Construction, many new quantum codes could be obtained from these codes, which are not covered in the literature.

New quantum codes derived from a family of antiprimitive BCH codes

2017 ◽  
Vol 15 (07) ◽  
pp. 1750052 ◽  
Author(s):  
Yang Liu ◽  
Ruihu Li ◽  
Liangdong Lü ◽  
Luobin Guo
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Detailed Analysis ◽  
Communication System ◽  
Quantum Information Theory ◽  
Bch Codes ◽  
Quantum Codes ◽  
Narrow Sense ◽  
Classical Communication ◽  
Cyclotomic Cosets

The Bose–Chaudhuri–Hocquenghem (BCH) codes have been studied for more than 57 years and have found wide application in classical communication system and quantum information theory. In this paper, we study the construction of quantum codes from a family of [Formula: see text]-ary BCH codes with length [Formula: see text] (also called antiprimitive BCH codes in the literature), where [Formula: see text] is a power of 2 and [Formula: see text]. By a detailed analysis of some useful properties about [Formula: see text]-ary cyclotomic cosets modulo [Formula: see text], Hermitian dual-containing conditions for a family of non-narrow-sense antiprimitive BCH codes are presented, which are similar to those of [Formula: see text]-ary primitive BCH codes. Consequently, via Hermitian Construction, a family of new quantum codes can be derived from these dual-containing BCH codes. Some of these new antiprimitive quantum BCH codes are comparable with those derived from primitive BCH codes.

Construction of quantum negacyclic BCH codes

2018 ◽  
Vol 16 (07) ◽  
pp. 1850059 ◽  
Author(s):  
Xiaoshan Kai ◽  
Ping Li ◽  
Shixin Zhu
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Bch Codes ◽  
Quantum Codes ◽  
Negacyclic Codes ◽  
Cyclotomic Cosets ◽  
Better Than

Let [Formula: see text] be an odd prime power and [Formula: see text] be a positive integer. Maximum designed distance such that negacyclic BCH codes over [Formula: see text] of length [Formula: see text] are Hermitian dual-containing codes is given. The dimension of such Hermitian dual-containing negacyclic codes is completely determined by analyzing cyclotomic cosets. Quantum negacyclic BCH codes of length [Formula: see text] are obtained by using Hermitian construction. The constructed quantum negacyclic BCH codes produce new quantum codes with parameters better than those obtained from quantum BCH codes.

QUANTUM CODES FROM CYCLIC CODES OVER FINITE RING

2009 ◽  
Vol 07 (06) ◽  
pp. 1277-1283 ◽  
Author(s):  
JIANFA QIAN ◽  
WENPING MA ◽  
WANGMEI GUO
Keyword(s):  
Finite Field ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Finite Ring ◽  
New Method ◽  
Quantum Codes ◽  
Orthogonal Codes ◽  
Quantum Error

A new method to obtain self-orthogonal codes over finite field F2 is presented. Based on this method, we provide a construction for quantum error-correcting codes starting from cyclic codes over finite ring R = F2 + uF2. As an example, we present infinite families of quantum error-correcting codes which are derived from cyclic codes over the ring R = F2 + uF2.

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