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On the b-Symbol Distances of matrix product codes, Constacyclic codes, and Reed-Muller codes

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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matrix product codes over finite commutative Frobenius rings

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DESIGNS codeS AND CRYPTOGRAPHY 2014年第2期71卷 201-227页

作者： Fan, Yun Ling, San Liu, Hongwei Cent China Normal Univ Sch Math & Stat Wuhan 430079 Peoples R China Nanyang Technol Univ Sch Phys Math Sci Div Math Sci Singapore 637616 Singapore

Properties of matrix product codes over finite commutative Frobenius rings are investigated. The minimum distance of matrix product codes constructed with several types of matrices is bounded in different ways. The du... 详细信息

关键词： matrix product code Frobenius ring Minimum distance Dual code Quasi-cyclic code

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matrix product codeS WITH ROSENBLOOM-TSFASMAN METRIC

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Acta Mathematica Scientia 2013年第3期33卷 687-700页

作者：陈博聪林丽仁刘宏伟 School of Mathematics and Statistics Central China Normal University

In this article, the Rosenbloom-Tsfasman metric of matrix product codes over finite commutative rings is studied and the lower bounds for the minimal Rosenbloom- Tsfasman distances of the matrix product codes axe obtained. The lower bounds of the dual codes of matrix product codes over finite commutative Frobenius rings are also given.

关键词： Finite commutative Frobenius ring matrix product code Rosenbloom-Tsfasman metric

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Homogeneous weights of matrix product codes over finite principal ideal rings

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FINITE FIELDS AND THEIR APPLICATIONS 2014年第Sep.期29卷 247-267页

作者： Fan, Yun Ling, San Liu, Hongwei Cent China Normal Univ Sch Math & Stat Wuhan 430079 Peoples R China Nanyang Technol Univ Sch Phys & Math Sci Div Math Sci Singapore 637616 Singapore

In this paper, the homogeneous weights of matrix product codes over finite principal ideal rings are studied and a lower bound for the minimum homogeneous weights of such matrix product codes is obtained. (C) 2014 Els... 详细信息

关键词： Finite principal ideal ring Homogeneous weight matrix product code

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Some self-dual codes and isodual codes constructed by matrix product codes

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DESIGNS codeS AND CRYPTOGRAPHY 2024年第11期92卷 3533-3559页

作者： Pan, Xu Chen, Hao Liu, Hongwei Jinan Univ Sch Informat Sci & Technol Cyber Secur Guangzhou 510632 Guangdong Peoples R China Cent China Normal Univ Sch Math & Stat Wuhan 430079 Hubei Peoples R China

In 2020, Cao et al. proved that any repeated-root constacyclic code is monomially equivalent to a matrix product code of simple-root constacyclic codes. In this paper, we study a family of matrix product codes with wonderful properties, which is a generalization of linear codes obtained from the [u+v|u-v]-construction and [u+v|lambda(-1)u-lambda(-1)v]-construction. Then we show that any lambda-constacyclic code (not necessary repeated-root lambda-constacyclic code) of length N over the finite field F-q with gcd(q-1/ord(lambda), N) >= 2, where ord(lambda) is the order of lambda in the cyclic group F-q* = F-q\matrix product code, is a matrix product code of some constacyclic codes. It is a highly interesting question that the existence of sequences {C-1, C-2, C-3, ...} of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances, i.e., C-i is an [n(C-i), k(Ci), d(C-i)](q)-linear code such that lim(i ->+infinity) n(C-i) = +infinity and lim(i ->+infinity)d(C-i)/root n(C-i) > 0. Based on the [u+v|lambda(-1)u-lambda(-1)v]-construction, we construct several families of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances by using Reed-Muller codes, projective Reed-Muller codes. And we construct some new Euclidean isodual lambda-constacyclic codes with square-root-like minimum Hamming distances from Euclidean self-dual cyclic codes and Euclidean self-dual negacyclic codes by monomial equivalences.

关键词： matrix product code Constacyclic code Self-dual code Isodual code

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Good Entanglement-Assisted Qubit codes from matrix product codes

Good Entanglement-Assisted Qubit Codes from Matrix Product C...

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2024 Information Theory Workshop

作者： Sok, Lin Ezerman, Martianus Frederic Ling, San Nanyang Technol Univ Sch Phys & Math Sci 21 Nanyang Link Singapore 637371 Singapore

ISBN: (纸本)9798350348941;9798350348934

We study the Hermitian hulls of matrix product codes and use them to design linear codes with arbitrary hull dimensions. We continue by looking into some propagation rules that preserve the hull dimensions of F4-linear codes. We propose a recursive method to keep the Hermitian hull dimension fixed while increasing the dimensions or the minimum distances of codes built from a given F4-linear code. Using the Hermitian construction route, we derive good entanglement-assisted quantum codes from matrix product codes and recursively iterated codes.

关键词： entanglement-assisted quantum code Hermitian hull matrix product code propagation rule

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Linear codes with arbitrary dimensional hull and their applications to EAQECCs

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QUANTUM INFORMATION PROCESSING 2022年第2期21卷 1-30页

作者： Sok, Lin Qian, Gang Anhui Univ Sch Math Sci Hefei 230601 Peoples R China

The Euclidean hull dimension of a linear code is an important quantity to determine the parameters of an entanglement-assisted quantum error-correcting code (EAQECC) if the Euclidean construction is applied. In this paper, we study the Euclidean hull of a linear code by means of orthogonal matrices. We provide some methods to construct linear codes over F-pm with hull of arbitrary dimensions. With existence of self-dual bases of F-pm over F-p, we determine a Gray map from F-pm to F-p(m), and from a given linear code over F-pm with one-dimensional hull, we construct, using such a Gray map, a linear code over F-p with m-dimensional hull for all m when p is even and for all m odd when p is odd. Comparisons with classical constructions are made, and some good EAQECCs over F-q, q = 2, 3, 4, 5, 9, 13, 17, 49 are presented.

关键词： Hull Optimal code Orthogonal matrix matrix product code Self-dual basis Gray map Entanglement-assisted quantum error-correcting code

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Galois hulls of linear codes over finite fields

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DESIGNS codeS AND CRYPTOGRAPHY 2020年第2期88卷 241-255页

作者： Liu, Hongwei Pan, Xu Cent China Normal Univ Sch Math & Stat Wuhan 430079 Peoples R China

The l-Galois hull h(l)(C) of an [n, k] linear code C over the finite field F-q is the intersection of C and C-perpendicular to l, where C-perpendicular to l denotes the l-Galois dual of C which was introduced by Fan and Zhang in 2017. The l-Galois LCD code is a linear code C satisfying h(l)(C) = C boolean AND C-perpendicular to l = matrix product code. In this paper, we show that the dimension of the l-Galois hull of a linear code is invariant under permutation equivalences and we provide a method to calculate the dimension of the l-Galois hull through a generator matrix of the code. Moreover, we obtain that the dimension of the l-Galois hull of a ternary code is also invariant under monomial equivalences. We show that every [n, k] linear code over Fq is monomial equivalent to an l-Galois LCD code for any q > 4. We conclude that if there exists an [n, k] linear code over Fq for any q > 4, then there exists an l-Galois LCD code with the same parameters for any 0 <= l <= e - 1, where q = p(e) for some prime number p. As an application, we characterize the l-Galois hulls of matrix product codes over finite fields.

关键词： l-Galois hull of a linear code Monomial equivalence l-Galois LCD code matrix product code

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matrix product and quasi-twisted codes in one class

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第2期43卷 1-20页

作者： Eldin, Ramy Taki Ain Shams Univ Fac Engn Cairo Egypt Egypt Univ Informat Knowledge City New Adm Capital Egypt

Many classical constructions, such as Plotkin's and Turyn's, were generalized by matrix product (MP) codes. Quasi-twisted (QT) codes, on the other hand, form an algebraically rich structure class that contains many codes with best-known parameters. We significantly extend the definition of MP codes to establish a broader class of generalized matrix product (GMP) codes that contains QT codes as well. We propose a generator matrix formula for any linear GMP code and provide a condition for determining the code size. We prove that any QT code has a GMP structure. Then we show how to build a generator polynomial matrix for a QT code from its GMP structure, and vice versa. Even though the class of QT codes contains many codes with best-known parameters, we present different examples of GMP codes with best-known parameters that are neither MP nor QT. Two different lower bounds on the minimum distance of GMP codes are presented;they generalize their counterparts in the MP codes literature. The second proposed lower bound replaces the non-singular by columns matrix with a less restrictive condition. Some examples are provided for comparing the two proposed bounds, as well as showing that these bounds are tight.

关键词： matrix product code Quasi-twisted code Best-known parameter Lower bound

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LCD matrix product codes with an application

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DISCRETE MATHEMATICS ALGORITHMS AND APPLICATIONS 2023年第4期15卷

作者： Bhowmick, Sanjit Bagchi, Satya Bandi, Ramakrishna Natl Inst Technol Durgapur Dept Math Durgapur 713209 India Dr SPM Int Inst Informat Technol Ctr Appl Math Naya Raipur India

matrix product (MP) codes over a finite field F-q, where q a prime power, were first introduced by Blackmore and Norton [MP codes over F-q, Appl. Algebra Engrg. Comm. Comput. 12 (2001) 477-500]. This paper investigates linear complementary dual (LCD) MP codes over finite commutative Frobenius rings. We derive a necessary and sufficient condition for an MP code to have an LCD. We provide some efficient constructions of LCD MP codes. Finally, we have shown an application on cryptography of LCD codes.

关键词： matrix product code Frobenius ring LCD code

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