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The dual code of any (δ plus αu²)-constacyclic code over F2m[u]/⟨u⁴⟩ of oddly even length

DISCRETE MATHEMATICS 2019年第8期342卷 2179-2191页

作者： Gao, Yun Fu, Fang-Wei Nankai Univ Chern Inst Math Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China

Let F-2m be a finite field of cardinality 2(m), R = F-2m[u]/ = F-2m + uF(2m) +u(2)F(2m) + u(3)F(2m) (u(4) = 0) which is a finite chain ring, and n is an odd positive integer. For any delta, alpha is an element of F-2m... 详细信息

Let F-2m be a finite field of cardinality 2(m), R = F-2m[u]/< u(4)> = F-2m + uF(2m) +u(2)F(2m) + u(3)F(2m) (u(4) = 0) which is a finite chain ring, and n is an odd positive integer. For any delta, alpha is an element of F-2m(x) an explicit representation for the dual code of any (delta + alpha u(2))-constacyclic code over R of length 2n is given. And some dual codes of (1 + u(2))-constacyclic codes over R of length 14 are constructed. For the case of delta = 1, all distinct self-dual (1 + alpha u(2))-constacyclic codes over R of length 2n are determined. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Dual code Self-dual code constacyclic code

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A tight upper bound on the number of non-zero weights of a constacyclic code

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FINITE FIELDS AND THEIR APPLICATIONS 2024年 93卷

作者： Zhang, Hanglong Cao, Xiwang Nanjing Univ Aeronaut & Astronaut Dept Math Nanjing 210016 Peoples R China MIIT Key Lab Math Modelling & High Performance Comp Air Nanjing 211106 Peoples R China

For a simple-root lambda-constacyclic code Cover F-q, let and be the subgroups of the automorphism group of C generated by the cyclic shift rho, and by the cyclic shift rho and the scalar multiplication group rho, respectively. Let N-G(C*) be the number of orbits of a subgroup Gof the automorphism group of C acting on C* = C\constacyclic code. In this paper, we establish explicit formulas for N- (C*) and N- (C*). Consequently, we derive a upper bound on the number of non-zero weights of C. We present some irreducible and reducible lambda-constacyclic codes, which show that the upper bound is tight. A sufficient condition to guarantee N- (C*) = N- (C*) is presented. (c) 2023 Elsevier Inc. All rights reserved.

关键词： Cyclic code constacyclic code Hamming weight Group action

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Torsion codes of a α-constacyclic code over Fpm[u]/⟨u³⟩

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DISCRETE MATHEMATICS 2024年第6期347卷

作者： Hesari, Roghaye Mohammadi Samei, Karim Bu Ali Sina Univ Fac Sci Dept Math Hamadan Iran

Let F-pm be a finite field of cardinality p(m) and R-3= F-pm[u]/, where p is a prime, mis a positive integer and u(3)= 0. Laaouine et al. (2021) [7] obtained the torsion code and the number of codewords of cyclic code... 详细信息

Let F-pm be a finite field of cardinality p(m) and R-3= F-pm[u]/< u(3)>, where p is a prime, mis a positive integer and u(3)= 0. Laaouine et al. (2021) [7] obtained the torsion code and the number of codewords of cyclic codes of length p(s) over F(p)m[u]/< u(3)>, but some of the results were wrong, which in this paper, we improve them. By a new method, we compute the number of codewords of these codes. Finally, by using a one-to-one correspondence between cyclic and alpha-constacyclic codes of length psover F-pm[u]/< u(3)>, we generalize these results to alpha-constacyclic codes, where a is a non-zero element of the finite field F(p)m. (c) 2024 Published by Elsevier B.V.

关键词： Chain ring constacyclic code Local ring Torsion

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New Quantum MDS code from constacyclic codes

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Chinese Annals of Mathematics,Series B 2016年第6期37卷 891-898页

作者： Liqin HU Qin YUE Xiaomeng ZHU School of Communication Engineering Hangzhou Dianzi University Hangzhou 310018 China Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing 210016China

In recent years, there have been intensive activities in the area of constructing quantum maximum distance separable(MDS for short) codes from constacyclic MDS codes through the Hermitian construction. In this paper, a new class of quantum MDS code is constructed, which extends the result of [Theorems 3.14–3.15, Kai, X., Zhu, S., and Li,P., IEEE Trans. on Inf. Theory, 60(4), 2014, 2080–2086], in the sense that our quantum MDS code has bigger minimum distance.

关键词： Cyclotomic coset Quantum MDS code constacyclic code BCH bound

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Repeated-root constacyclic codes of length kslmpn over finite fields

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FINITE FIELDS AND THEIR APPLICATIONS 2025年 101卷

作者： Zhang, Qi Wang, Weiqiong Luo, Shuyu Li, Yue Changan Univ Sch Sci Xian 710064 Peoples R China Southeast Univ Natl Mobile Commun Res Lab Nanjing 210096 Peoples R China

For different odd primes k , l , p , and positive integers s , m , n , the polynomial x k s l m p n - lambda in F q [x] is explicitly factorized, where p is the characteristic of F q , lambda is an element of F q & lowast;. All repeated-root constacyclic codes and their dual codes of length k s l m p n over F q are characterized. In addition, the characterization and enumeration of all linear complementary dual (LCD) cyclic and negacyclic codes of length k s l m p n over F q are obtained. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Finite field constacyclic code Generator polynomial Dual code LCD code

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Equivalence of constacyclic codes with shift constants of different orders

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DESIGNS codeS AND CRYPTOGRAPHY 2025年第1期93卷 79-93页

作者： Dastbasteh, Reza Padashnick, Farzad Crespo, Pedro M. Grassl, Markus Sharafi, Javad Univ Navarra Tecnun Donostia San Sebastian Spain Univ Isfahan Dept Pure Math Esfahan Iran Univ Gdansk Int Ctr Theory Quantum Technol Gdansk Poland Kashan Univ Kashan Iran

Let a and b be two non-zero elements of a finite field F-q, where q>2. It has been shown that if a and b have the same multiplicative order in F-q, then the families of a-constacyclic and b-constacyclic codes over F-q are monomially equivalent. In this paper, we investigate the monomial equivalence of a-constacyclic and b-constacyclic codes when a and b have distinct multiplicative orders. We present novel conditions for establishing monomial equivalence in such constacyclic codes, surpassing previous methods of determining monomially equivalent constacyclic and cyclic codes. As an application, we use these results to search for new linear codes more systematically. In particular, we present more than 70 new record-breaking linear codes over various finite fields, as well as new binary quantum codes.

关键词： constacyclic code Cyclic code Monomial equivalence Isometric equivalence New linear code

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On the b-Symbol Distances of Matrix Product codes, constacyclic codes, and Reed-Muller codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2025年第1期71卷 287-296页

作者： Pan, Xu Ling, San Liu, Hongwei Chen, Bocong Jinan Univ Sch Informat Sci & Technol Cyber Secur Guangzhou 510632 Guangdong Peoples R China Nanyang Technol Univ Sch Phys & Math Sci Singapore 637371 Singapore Cent China Normal Univ Sch Math & Stat Key Lab NAA MOE Wuhan 430079 Peoples R China South China Univ Technol Sch Math Guangzhou 510641 Guangdong Peoples R China

Matrix product codes are generalizations of some well-known classes of codes, including generalized Reed-Muller codes and repeated-root constacyclic codes. Recently, a bound for the minimum symbol-pair distance of a matrix product code was given by (Luo et al., 2023), leading to the creation of new families of MDS symbol-pair codes. In this paper, we provide lower and upper bounds for the minimum b-symbol distance of matrix product codes, which naturally extends some of the results by (Luo et al., 2023). Examples meeting the bounds are included to illustrate our results. As an initial application of these new bounds, we establish that constacyclic codes (repeated-root or otherwise) meeting specific criteria can indeed be classified as matrix product codes, and present bounds on the minimum b-symbol distance of such constacyclic codes. Additionally, we determine all the minimum b-symbol distances of Reed-Muller codes as a secondary application of the new bounds.

关键词： codes Product codes Linear codes Reed-Muller codes Symbols Vectors Upper bound Hamming weight Generators Mathematics Matrix product code constacyclic code Reed-Muller code b-symbol distance

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constacyclic codes over Z2 [u]/⟨u²⟩ x Z2 [u]/⟨u³⟩ and the MacWilliams identities

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2024年 1-30页

作者： Sagar, Vidya Yadav, Ankit Sarma, Ritumoni Indian Inst Technol Delhi Dept Math New Delhi 110016 Delhi India

In this article, we deal with additive codes over the Frobenius ring R2R3 := Z(2)[u]/ x Z(2)[u]/. First, we study constacyclic codes over R-2 and R-3 and find their generator polynomials. With the help of these genera... 详细信息

In this article, we deal with additive codes over the Frobenius ring R2R3 := Z(2)[u]/< u(2)> x Z(2)[u]/< u(3)>. First, we study constacyclic codes over R-2 and R-3 and find their generator polynomials. With the help of these generator polynomials, we determine the structure of constacyclic codes over R2R3. We use Gray maps to show that constacyclic codes over R2R3 are essentially binary generalized quasi-cyclic codes. Moreover, we obtain a number of binary codes with good parameters from these R2R3-constacyclic codes. Besides, several weight enumerators are computed, and the corresponding MacWilliams identities are established.

关键词： constacyclic code Gray map Quasi-cyclic code Weight enumerator MacWilliams identities

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constacyclic codes over finite fields

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FINITE FIELDS AND THEIR APPLICATIONS 2012年第6期18卷 1217-1231页

作者： Chen, Bocong Fan, Yun Lin, Liren Liu, Hongwei Cent China Normal Univ Sch Math & Stat Wuhan 430079 Hubei Peoples R China

An equivalence relation called isometry is introduced to classify constacyclic codes over a finite field: the polynomial generators of constacyclic codes of length l(t)p(s) are characterized, where p is the characteri... 详细信息

关键词： Finite field constacyclic code Isometry Polynomial generator

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On the Gray Images of Some constacyclic codes over Fp ＋ uFp ＋ u^2Fp

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Journal of Systems Science & Complexity 2016年第3期29卷 842-849页

作者： YU Haifeng ZHU Shixin KAI Xiaoshan Department of Mathematics and Physics Hefei University Hefei 230601 China. School of Mathematics Hefei University of Technology Hefei 230009 China National Mobile Communications Research Laboratory Southeast University Nanjinq 210096 China.

Abstract constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, a new Gray map between codes over Fp ＋ uFp ＋ u^2Fp and codes over Fp is defined, where p is an odd prime. By means of this map, it is shown that the Gray image of a linear （1 ＋ u ＋ u2）-constacyclic code over Fp ＋ uFp ＋ u^2Fp of length n is a repeated-root cyclic code over Fp of length pn. Furthermore, some examples of optimal linear cyclic codes over F3 from （1 ＋ u ＋ u2）-constacyclic codes over F3 ＋ uF3 ＋ u^2F3 are given.

关键词： constacyclic code cyclic code Gray map.

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