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negacyclic dually-BCH codes of length qm-1

CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2025年 1-15页

作者： Yu, Ming Kai, Xiaoshan Hefei Univ Technol Sch Math Hefei 230601 Anhui Peoples R China

In this paper, we investigate negacyclic dually-BCH codes over GF(q) of length n = qm-1. We provide the first three largest odd coset leaders modulo 2n. Sufficient and necessary conditions in terms of designed distanc... 详细信息

关键词： negacyclic code BCH code Cyclic code

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The dual codes of negacyclic BCH codes of length qm+1/2

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DISCRETE MATHEMATICS 2025年第9期348卷

作者： Fu, Yuqing Liu, Hongwei Zhoukou Normal Univ Sch Math & Stat Zhoukou 466001 Peoples R China Cent China Normal Univ Sch Math & Stat Key Lab NAA MOE Wuhan 430079 Peoples R China

negacyclic BCH codes form an important subclass of negacyclic codes and can produce optimal linear codes in many cases. The question of whether the dual code of a negacyclic BCH code is a negacyclic BCH code is, in general, very hard to answer. To investigate further the properties of the dual codes of negacyclic BCH codes, the concept of negacyclic duallyBCH codes is proposed in this paper and then the dual codes of narrow-sense negacyclic BCH codes of length qm+1 2 over the finite field Fq are studied, where q = 3 (mod 4). Some lower bounds on the minimum distances of the dual codes are established, which are very close to the true minimum distances of the dual codes in many cases. Sufficient and necessary conditions in terms of designed distances are presented to ensure that narrow-sense negacyclic BCH codes of length qm+1 2 are negacyclic dually-BCH codes. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： BCH code Dually-BCH code negacyclic code Minimum distance

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negacyclic codes over the local ring Z₄[v]/⟨v²+2v⟩ of oddly even length and their Gray images

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FINITE FIELDS AND THEIR APPLICATIONS 2018年第Jul.期52卷 67-93页

作者： Cao, Yuan Cao, Yonglin Shandong Univ Technol Sch Math & Stat Zibo 255091 Shandong Peoples R China

Let R = Z(4)[v]/ = Z(4) + vZ(4) (v(2) = 2v) and n be an odd positive integer. Then R is a local non-principal ideal ring of 16 elements and there is a Z(4)-linear Gray map from R onto Z(4)(2) which preserves Lee dista... 详细信息

Let R = Z(4)[v]/< v(2) + 2v > = Z(4) + vZ(4) (v(2) = 2v) and n be an odd positive integer. Then R is a local non-principal ideal ring of 16 elements and there is a Z(4)-linear Gray map from R onto Z(4)(2) which preserves Lee distance and orthogonality. First, a canonical form decomposition and the structure for any negacyclic code over R of length 2n are presented. From this decomposition, a complete classification of all these codes is obtained. Then the cardinality and the dual code for each of these codes are given, and self-dual negacyclic codes over R of 2 length 2n are presented. Moreover, all 23 . (4(P) + 5 . 2(P) + 9)2(p)-2/p negacyclic codes over R of length 2M(p) and all 3 . (4(P) + 5 . 2(P) + 2(p-1)-1 /9) self-dual codes among them are presented precisely, where M-P = 2(P) - 1 is a Mersenne prime. Finally, 36 new and good self-dual 2 -quasi-twisted linear codes over Z(4) with basic parameters (28, 2(28), d(L) = 8, d(E) = 12) and of type 2(14)4(7) and basic parameters (28, 2(28), d(L) = 6, d(E) = 12) and of type 21848 which are Gray images of self-dual negacyclic codes over R of length 14 are listed. (C) 2018 Elsevier Inc. All rights reserved.

关键词： negacyclic code Dual code Self-dual code Local ring Finite chain ring

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The Hamming Distances of a Class of p-Ary negacyclic codes

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Chinese Journal of Electronics 2018年第1期27卷 46-51页

作者： DING Jian LI Hongju Department of General Education Anhui Xinhua University

Based on the construction of a new distance-preserving Gray map from(（Fp+ uFp）N, Gray distance) to(the corresponding Gray images in FppN, Hamming distance) and the calculation of Gray distances of（u-1）-constacyclic codes over Fp+ uFp, a bound for the Hamming distances of a class of negacyclic codes with length p N over Fpis obtained, which is more tighter than Singleton bound. Further more, the exact Hamming distances of some p-ary negacyclic codes are determined from this bound, some of which cannot be got from Dinh's work published on Finite Fields and Their Applications in 2008.

关键词： negacyclic code Hamming distance Gray distance Gray map

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New quantum codes from negacyclic codes of length p^s over Fp^m + uFp^m

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QUANTUM INFORMATION PROCESSING 2023年第1期22卷 1-10页

作者： Boudine, Brahim Laaouine, Jamal Charkani, Mohammed Elhassani Sidi Mohamed Ben Abdellah Univ Fac Sci Dhar Mahraz Fes Morocco

Let p be a prime integer and m and s be two positive integers. We compute dual codes and Hamming distances of negacyclic codes of length p(s) over Fp(m) + uFp(m) with u(2) = 0. This yields new quantum codes from negac... 详细信息

关键词： Quantum codes negacyclic code Error-correcting codes

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Several Families of Ternary negacyclic codes and Their Duals

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IEEE TRANSACTIONS ON INFORMATION THEORY 2024年第5期70卷 3226-3241页

作者： Sun, Zhonghua Ding, Cunsheng Hefei Univ Technol Sch Math Hefei 230601 Anhui Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China

Constacyclic codes contain cyclic codes as a subclass and have nice algebraic structures. Constacyclic codes have theoretical importance, as they are connected to a number of areas of mathematics and outperform cyclic codes in several aspects. negacyclic codes are a subclass of constacyclic codes and are distance-optimal in many cases. However, compared with the extensive study of cyclic codes, negacyclic codes are much less studied. In this paper, several families of ternary negacyclic codes and their duals are constructed and analysed. These families of negacyclic codes and their duals contain distance-optimal codes and have very good parameters in general. The duals of three families of ternary negacyclic codes presented in this paper are distance-optimal.

关键词： codes Linear codes Generators Sun Mathematics codecs Measurement uncertainty Cyclic code negacyclic code linear code

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Entanglement-assisted quantum codes from cyclic codes and negacyclic codes

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QUANTUM INFORMATION PROCESSING 2020年第5期19卷 138-138页

作者： Wang, Junli Li, Ruihu Lv, Jingjie Song, Hao Air Force Engn Univ Dept Basic Sci Xian 710051 Shaanxi Peoples R China

Entanglement-assisted quantum error-correcting (EAQEC) codes could generalize and improve performance of standard quantum error-correcting (QEC) codes to a great extent. In this paper, series of EAQEC codes of length n=q-1a(q+1) are constructed from cyclic codes and negacyclic codes, where q is a prime power and a is a positive integer such that a divide (q-1). It turns out that the number of required entanglement bits can take almost all possible values. Consequently, our EAQEC codes have flexible parameters and most of them are new. For given the same length, our construction contain and extend those known consequences in Grassl et al. (Int J Quantum Inf 2(1):55-64, 2004), Jin et al. (IEEE Trans Inf Theory 56:4735-4740, 2010), Kai et al. (IEEE Trans Inf Theory 60:2080-2086, 2014), Jin and Xing (IEEE Trans Inf Theory 60:2921-2925, 2014), Chen et al. (IEEE Trans Inf Theory 61:1474-1484, 2015), Zhang and Ge (IEEE Trans Inf Theory 61:5224-5228, 2015;Des codes Cryptogr 83(3):503-517, 2016), Shi et al. (Cryptogr Commun 10(6):1165-1182, 2018;Finite Fields Appl 46:347-362, 2017), Fan et al. (Quantum Inf Comput 16:423-434, 2016), Lu et al. (Quantum Inf Process 17(69):1-23, 2018), Li et al. (Int J Quantum Inf 17(1):1950022, 2019), Liu et al. (Quantum Inf Process 17(210):1-19, 2018), Fang et al. (Euclidean and Hermitian Hulls of MDS codes and Their Applications to EAQECCs. arXiv:). Above all, all our codes are maximum-distance-separable (MDS) if their minimum distance d <= n+22.

关键词： Cyclic code negacyclic code Cyclotomic coset EAQEC code

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Self-Dual negacyclic codes With Variable Lengths and Square-Root-Like Lower Bounds on the Minimum Distances

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IEEE TRANSACTIONS ON INFORMATION THEORY 2024年第7期70卷 4879-4888页

作者： Xie, Conghui Chen, Hao Ding, Cunsheng Sun, Zhonghua Jinan Univ Coll Informat Sci & Technol Cyber Secur Guangzhou 510632 Guangdong Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China Hefei Univ Technol Sch Math Hefei 230601 Anhui Peoples R China

The construction of self-dual codes with large minimum distances has been an active topic in coding theory. The construction and classification of extremal self-dual codes over small fields are related to other fields of mathematics, such as lattices and invariant theory as well as combinatorial t -designs. It is well-known that q -ary self-dual cyclic codes exist only when q is an even prime power and q -ary self-dual negacyclic codes exist for any odd prime power q . In 2009 a family of binary self-dual cyclic codes with lengths n(i) and minimum distances d(i )>= 1/2 root n(i), where ni goes to the infinity if i goes to the infinity, was constructed. In this paper, we construct several families of q -ary self-dual negacyclic codes of lengths n with their minimum distances larger than or equal to n(1/2) for various lengths n and any given odd prime power q . When q is an element of {3,5} and the length is small, the minimum distances of the constructed self-dual negacyclic codes are comparable with these self-dual codes with largest known minimum distances in the literature.

关键词： codes Linear codes Liquid crystal displays Sun Hamming distances Vectors Upper bound Self-dual code negacyclic code BCH bound

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Study on negacyclic codes over the ring Zp[u]/ <u^k+1 - u >

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2019年第1-2期59卷 693-700页

作者： Bag, Tushar Upadhyay, Ashish K. Indian Inst Technol Patna Dept Math Patna 801103 Bihar India

In this paper, we study the properties of negacyclic codes over the ring R = Z(p) + uZ(p) + ... + u(k)Z(p), for odd prime p, using decomposition method. We have determined the generators of negacyclic and dual negacyclic codes. We have also established the necessary and sufficient condition for it to contains it's dual. It is also shown that the Z(p)-Gray image of a negacyclic code of length n is a quasi negacyclic code of length 3n.

关键词： negacyclic code Quasi negacyclic code Gray map Dual code

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Infinite Families of Cyclic and negacyclic codes Supporting 3-Designs

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IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第4期69卷 2341-2354页

作者： Wang, Xiaoqiang Tang, Chunming Ding, Cunsheng Hubei Univ Fac Math & Stat Hubei Key Lab Appl Math Wuhan 430062 Peoples R China China West Normal Univ Sch Math & Informat Nanchong 637002 Sichuan Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China

Interplay between coding theory and combinatorial t-designs has been a hot topic for many years for combinatori-alists and coding theorists. Some infinite families of cyclic codes supporting infinite families of 3-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of 3-designs has been reported in the literature. This is the main motivation of this paper. Let q = p(m), where p is an odd prime and m = 2 is an integer. The objective of this paper is to present an infinite family of cyclic codes over GF(q) supporting an infinite family of 3-designs and two infinite families of negacyclic codes over GF(q(2)) supporting two infinite families of 3-designs. The parameters and the weight distributions of these codes are determined. The subfield subcodes of these negacyclic codes over GF(q) are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over GF(4) supporting a 4-design and seven open problems are also presented in this paper.

关键词： codes Linear codes Mathematics Hamming weight Indexes Hamming distances Generators Constacyclic code cyclic code negacyclic code t-design Steiner system

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