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additive twisted codes: new distance bounds and infinite families of quantum codes

DESIGNS codeS AND CRYPTOGRAPHY 2025年 1-38页

作者： Dastbasteh, Reza Lisonek, Petr Tecnun Univ Navarra Donostia San Sebastian Spain Simon Fraser Univ Dept Math Burnaby BC Canada

We provide a new construction of quantum codes that enables integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Next, we present new connections between twisted codes and linear cyclic codes and provide novel bounds for the minimum distance of twisted codes. We show that classical tools such as the Hartmann-Tzeng minimum distance bound are applicable to twisted codes. This enabled us to discover five new infinite families and many other examples of record-breaking, and sometimes optimal, binary quantum codes. We also discuss the role of the gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} value on the parameters of twisted codes and present new results regarding the construction of twisted codes with different gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} values but identical parameters. Finally, we list many new record-breaking binary quantum codes that we obtained from additive twisted, linear cyclic, and constacyclic codes.

关键词： additive code Twisted code Cyclic code Quantum code Minimum distance bound

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ACD codes and cyclic codes over Z2Rk

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COMPUTATIONAL & APPLIED MATHEMATICS 2025年第1期44卷 1-17页

作者： Yadav, Ankit Sagar, Vidya Sarma, Ritumoni Indian Inst Technol Delhi Dept Math New Delhi 110016 India Indian Inst Sci Dept Elect Syst Engn Bengaluru 560012 Karnataka India

We deal with ACD (additive complementary dual) codes and cyclic codes over the mixed alphabet Z2Rk, where Rk := Z2[ y]/yk , k = 2. First, we establish a few criteria for Z2Rk -additive codes to be ACD codes. We also present conditions for separable codes and a class of additive codes (not necessarily separable) over Z2Rk to be ACD codes that are both necessary and sufficient. With the help of a Gray map, binary LCD codes are obtained from Z2Rk -additive codes. Moreover, we describe the generator polynomial of the dual of an additive cyclic code over Z2Rk. Finally, we construct examples of optimal binary codes as the Gray image of certain additive cyclic codes over Z2Rk.

关键词： Cyclic code Self-dual code additive code LCD code ACD code

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additive codes over Galois rings

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FINITE FIELDS AND THEIR APPLICATIONS 2019年 56卷 332-350页

作者： Mahmoudi, Saadoun Samei, Karim Bu Ali Sina Univ Dept Math Hamadan Iran

Let S = GR(p(e), m) be a Galois ring of characteristic p(e) and cardinality p(em). An additive code over S of length n is a subgroup of S-n under addition. In this paper, we study additive codes over S. We introduce a correspondence between linear codes over Z(pe) and additive codes over S and we describe additive codes over S by the structure of linear codes over Z(pe). In particular, we find the generator matrix and the number of additive codes over S, and we determine some classes of MDR additive codes over S. Among other results, permutation equivalent additive codes and decomposable additive codes are described. Also we prove MacWilliams identity and Delsarte theorem for additive codes over S. (C) 2018 Published by Elsevier Inc.

关键词： additive code MDR code Generator matrix

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One and Two-Weight Z2R2 additive codes

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Chinese Journal of Electronics 2021年第1期30卷 72-76页

作者： LI Tiantian SHI Minjia LIN Bo WU Wenting School of Mathematical Sciences Anhui University School of Wendian Anhui University

This paper is devoted to the construction of one and two-weight Z2R2additive codes, where R2=F2[v]/（v4）. It is a generalization towards another direction of Z2Z4codes（S.T. Dougherty, H.W. Liu and L. Yu,"One weight Z2Z4additive codes", Applicable Algebra in Engineering, Communication and Computing, Vol.27,No.2, pp.123–138, 2016）. A Mac Williams identity which connects the weight enumerator of an additive code over Z2R2and its dual is established. Several construction methods of one-weight and two-weight additive codes over Z2R2are presented. Several examples are presented to illustrate our main results and some open problems are also proposed.

关键词： dual codes additive code weight enumerator two-weight additive codes

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Designs in additive codes over GF(4)

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DESIGNS codeS AND CRYPTOGRAPHY 2003年第2期30卷 187-199页

作者： Kim, JL Pless, V Univ Illinois Dept Math Stat & Comp Sci Chicago IL 60607 USA Univ Nebraska Dept Math Lincoln NE 68588 USA

The purpose of this paper is to study designs in additive codes over GF(4). There are two types of designs. One is a classical t-design with repeated blocks. In this case we have an analog of the Assmus Mattson Theorem for additive codes over GF(4). The other is a generalized t-design first introduced by Delsarte [4]. As an example, we consider the dodecacode, which is the unique additive even self-dual (12;2(12);6) code. We show that there exists a 5-(12, 6, 3) design in the dodecacode with either 3 distinct blocks or 3 repeated blocks covering a 5- set. We also find a new simple 3-(11, 5, 4) design in the shortened dodecacode by a computer search. Additionally we show that any extremal additive even self-dual code over GF(4) is homogeneous.

关键词： additive code Assmus-Mattson theorem generalized t-design

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On the Generator Polynomials of Skew Qusic-Cyclic codes over the Ring Z2Z4[u] 25

On the Generator Polynomials of Skew Qusic-Cyclic Codes over...

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Proceedings of the 2025 4th International Conference on Cryptography, Network Security and Communication Technology

作者： Huazhang Wu Jie Geng Jiahao Ding Runqiu Wu School of Mathematical Sciences Anhui University Hefei Anhui China

ISBN: (纸本)9798400712623

Encoding is an important research direction of information security. The coding theory has a lot of important applications in cryptogram, communication technology and network security, etc. Coding over a ring is one of important research objects in coding theory. In this article, we mainly study the problem for the generators for a special code called skew quasi-cyclic code over the ring ${Z}_2{Z}_4[u]$, where${u}^2 = 0$.

关键词： additive code

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On cyclic reversible self-dual additive codes with odd length over Ζ₂²

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IEEE TRANSACTIONS ON INFORMATION THEORY 2000年第3期46卷 1056-1059页

作者： Ran, M Snyders, J TelesciCOM Ltd IL-58810 Holon Israel Tel Aviv Univ Dept Elect Engn Syst IL-69978 Ramat Aviv Israel

Several additive codes of odd length over Z(2)(2) are introduced. These codes are cyclic and reversible. Furthermore, they are self-dual under an appropriately selected binary-valued inner product. Some binary derivatives of these codes have good parameters. Alt cyclic and reversible [5, 2.5, 3] additive codes over Z(2)(2) are isomorphic and possess interesting properties.

关键词： additive code cyclic code group code pentacode reversible code self-dual code

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Cyclic R -additive codes

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DISCRETE MATHEMATICS 2017年第7期340卷 1657-1668页

作者： Samei, Karim Mahmoudi, Saadoun Bu Ali Sina Univ Dept Math Hamadan Iran

Bierbrauer (2012) developed the theory of q-linear cyclic codes over (F-q)(m) and he obtained a parametric description of such codes by cyclotomic cosets. Recently, Cao et al. (2015) obtained the structure of cyclic additive codes over the Galois ring GR(p(e), m), where m is a prime integer. Let R be a finite commutative ring and R-n = R[x]/ < x(n) - 1 >. In this paper, we generalize the theory of F-q-linear codes over vector spaces to R-linear codes over free R-algebras (free as R-module). We call these codes, R-additive codes. We introduce a one-to-one correspondence between the classes of cyclic R-additive code and the classes of R-n-linear code. Using the structure of R-n-linear codes, we present the structure of cyclic R-additive codes, where R is a chain ring. Among other results, q-linear cyclic codes over (F-q)(m) are described by ring-theoretic facts, and the structure of cyclic additive codes over the Galois ring GR(p(e), m) is given for an arbitrary integer m, not necessarily a prime number. (c) 2016 Elsevier B.V. All rights reserved.

关键词： additive code Free R -algebra Chain ring Galois ring

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On the generators of Z2Z4[u]-additive codes and additive cyclic codes 24

On the generators of Z2Z4[u]-additive codes and additive cyc...

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3rd International Conference on Cryptography, Network Security and Communication Technology (CNSCT)

作者： Wu, Huazhang Geng, Jie Yu, Hao Anhui Univ Sch Math Sci Hefei 230601 Peoples R China

ISBN: (纸本)9798400716959

Encoding and decoding are two important directions of information security. Coding theory has so many important applications in cryptogram, communication technology and network security, etc. In this paper, a new type of additive cyclic codes over the ring Z(2)Z(4)[u] (u(2) = 0) in coding is introduced. We mainly focus on the study of the standard forms of the generator matrix and parity-check matrix of Z(2)Z(4)[u]-additive codes. Meanwhile, some algebraic properties of the generators for the dual ofZ(2)Z(4)[u]-additive cyclic codes are also discussed.

关键词： additive code additive cyclic code dual code generator matrix parity-check matrix

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The Nonexistence of a []-Quantum Stabilizer code

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IEEE TRANSACTIONS ON INFORMATION THEORY 2011年第7期57卷 4788-4793页

作者： Bierbrauer, Juergen Fears, Richard Marcugini, Stefano Pambianco, Fernanda Michigan Technol Univ Dept Math Sci Houghton MI 49931 USA Univ Perugia Dipartimento Matemat & Informat I-06100 Perugia Italy

One of the oldest problems in the theory of quantum stabilizer codes is solved by proving the nonexistence of quantum [[13, 5, 4]]-codes.

关键词： additive code codeline dimension hyperplane quantum code stabilizer code strength symplectic geometry weight

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