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Reversible $ G $-codes over the ring $ {mathcal{F}}_{j,k} $ with applications to DNA codes 可逆$ G $-环上$ {mathcal{F}}_{j,k} $的编码及其在DNA编码中的应用
IF 0.9 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-01-01 DOI: 10.3934/amc.2021056
Y. Cengellenmis, A. Dertli, S. Dougherty, Adrian Korban, S. Șahinkaya, Deniz Ustun

In this paper, we show that one can construct a begin{document}$ G $end{document}-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which begin{document}$ G $end{document}-codes are reversible of index begin{document}$ alpha $end{document}. Additionally, we introduce a new family of rings, begin{document}$ {mathcal{F}}_{j,k} $end{document}, whose base is the finite field of order begin{document}$ 4 $end{document} and study reversible begin{document}$ G $end{document}-codes over this family of rings. Moreover, we present some possible applications of reversible begin{document}$ G $end{document}-codes over begin{document}$ {mathcal{F}}_{j,k} $end{document} to reversible DNA codes. We construct many reversible begin{document}$ G $end{document}-codes over begin{document}$ {mathbb{F}}_4 $end{document} of which some are optimal. These codes can be used to obtain reversible DNA codes.

In this paper, we show that one can construct a begin{document}$ G $end{document}-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which begin{document}$ G $end{document}-codes are reversible of index begin{document}$ alpha $end{document}. Additionally, we introduce a new family of rings, begin{document}$ {mathcal{F}}_{j,k} $end{document}, whose base is the finite field of order begin{document}$ 4 $end{document} and study reversible begin{document}$ G $end{document}-codes over this family of rings. Moreover, we present some possible applications of reversible begin{document}$ G $end{document}-codes over begin{document}$ {mathcal{F}}_{j,k} $end{document} to reversible DNA codes. We construct many reversible begin{document}$ G $end{document}-codes over begin{document}$ {mathbb{F}}_4 $end{document} of which some are optimal. These codes can be used to obtain reversible DNA codes.
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引用次数: 3
Dualities for codes over finite Abelian groups 有限阿贝尔群上码的对偶性
IF 0.9 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-01-01 DOI: 10.3934/amc.2023023
S. Dougherty
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引用次数: 1
New dimension-independent upper bounds on linear insdel codes 线性内码的新维无关上界
IF 0.9 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-01-01 DOI: 10.3934/amc.2023008
Conghui Xie, Haoyuan Chen, Longjiang Qu, Ling Liu
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引用次数: 0
On $ mathbb{Z}_4mathbb{Z}_4[u^3] $-additive constacyclic codes 关于$ mathbb{Z}_4mathbb{Z}_4[u^3] $-加性常循环码
IF 0.9 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-01-01 DOI: 10.3934/amc.2022017
O. Prakash, S. Yadav, H. Islam, P. Solé

Let begin{document}$ mathbb{Z}_4 $end{document} be the ring of integers modulo begin{document}$ 4 $end{document}. This paper studies mixed alphabets begin{document}$ mathbb{Z}_4mathbb{Z}_4[u^3] $end{document}-additive cyclic and begin{document}$ lambda $end{document}-constacyclic codes for units begin{document}$ lambda = 1+2u^2,3+2u^2 $end{document}. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain begin{document}$ mathbb{Z}_4 $end{document}-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.

Let begin{document}$ mathbb{Z}_4 $end{document} be the ring of integers modulo begin{document}$ 4 $end{document}. This paper studies mixed alphabets begin{document}$ mathbb{Z}_4mathbb{Z}_4[u^3] $end{document}-additive cyclic and begin{document}$ lambda $end{document}-constacyclic codes for units begin{document}$ lambda = 1+2u^2,3+2u^2 $end{document}. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain begin{document}$ mathbb{Z}_4 $end{document}-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.
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引用次数: 1
Extremal absorbing sets in low-density parity-check codes 低密度奇偶校验码中的极值吸收集
IF 0.9 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-01-01 DOI: 10.3934/AMC.2021003
Emily McMillon, Allison Beemer, C. Kelley
Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of begin{document}$ b $end{document} for a given begin{document}$ a $end{document} for which an begin{document}$ (a,b) $end{document} -absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms.
Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of begin{document}$ b $end{document} for a given begin{document}$ a $end{document} for which an begin{document}$ (a,b) $end{document} -absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms.
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引用次数: 1
On recursive constructions of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-线性Hadamard码的递归构造
4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-01-01 DOI: 10.3934/amc.2023047
Dipak K. Bhunia, Cristina Fernández-Córdoba, Mercè Villanueva
The $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive codes are subgroups of $ mathbb{Z}_2^{alpha_1} times mathbb{Z}_4^{alpha_2} times mathbb{Z}_8^{alpha_3} $. A $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard code is a Hadamard code, which is the Gray map image of a $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive code. In this paper, we generalize some known results for $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard codes to $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes with $ alpha_1 neq 0 $, $ alpha_2 neq 0 $, and $ alpha_3 neq 0 $. First, we give a recursive construction of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive Hadamard codes of type $ (alpha_1, alpha_2, alpha_3;t_1, t_2, t_3) $ with $ t_1geq 1 $, $ t_2 geq 0 $, and $ t_3geq 1 $. It is known that each $ mathbb{Z}_4 $-linear Hadamard code is equivalent to a $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard code with $ alpha_1neq 0 $ and $ alpha_2neq 0 $. Unlike $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard codes, in general, this family of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes does not include the family of $ mathbb{Z}_4 $-linear or $ mathbb{Z}_8 $-linear Hadamard codes. We show that, for example, for length $ 2^{11} $, the constructed nonlinear $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-linear Hadamard codes are not equivalent to each other, nor to any $ mathbb{Z}_2 mathbb{Z}_4 $-linear Hadamard, nor to any previously constructed $ mathbb{Z}_{2^s} $-Hadamard code, with $ sgeq 2 $. Finally, we also present other recursive constructions of $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones.
The $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-加性代码是的子组 $ mathbb{Z}_2^{alpha_1} times mathbb{Z}_4^{alpha_2} times mathbb{Z}_8^{alpha_3} $. a $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $线性哈达玛码是一种哈达玛码,它是灰度图图像的一种 $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-附加代码。在本文中,我们推广了关于 $ mathbb{Z}_2 mathbb{Z}_4 $-线性Hadamard代码 $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-线性Hadamard代码与 $ alpha_1 neq 0 $, $ alpha_2 neq 0 $,和 $ alpha_3 neq 0 $. 首先,我们给出的递归构造 $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-加性Hadamard类型代码 $ (alpha_1, alpha_2, alpha_3;t_1, t_2, t_3) $ 有 $ t_1geq 1 $, $ t_2 geq 0 $,和 $ t_3geq 1 $. 众所周知,每一个 $ mathbb{Z}_4 $线性Hadamard代码相当于a $ mathbb{Z}_2 mathbb{Z}_4 $线性Hadamard代码 $ alpha_1neq 0 $ 和 $ alpha_2neq 0 $. 不像 $ mathbb{Z}_2 mathbb{Z}_4 $-线性哈达玛码,一般来说,这个族的 $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-线性Hadamard代码不包括族 $ mathbb{Z}_4 $-线性或 $ mathbb{Z}_8 $-线性Hadamard代码。例如,我们展示了长度 $ 2^{11} $,构造的非线性 $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $线性哈达玛码彼此不等价,也不等价于任何 $ mathbb{Z}_2 mathbb{Z}_4 $-线性Hadamard,也不是任何先前构建的 $ mathbb{Z}_{2^s} $-Hadamard code, with $ sgeq 2 $. 最后,我们也给出了的其他递归构造 $ mathbb{Z}_2 mathbb{Z}_4 mathbb{Z}_8 $-具有相同类型的加性Hadamard码,我们证明,应用Gray映射后,得到的码与之前的码是等价的。
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引用次数: 0
On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes 讨论了投影Reed-Muller码的最小距离、最小权码字和维数
4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-01-01 DOI: 10.3934/amc.2023035
Sudhir R. Ghorpade, Rati Ludhani
We give an alternative proof of the formula for the minimum distance of a projective Reed-Muller code of an arbitrary order. It leads to a complete characterization of the minimum weight codewords of a projective Reed-Muller code. This is then used to determine the number of minimum weight codewords of a projective Reed-Muller code. Various formulas for the dimension of a projective Reed-Muller code, and their equivalences are also discussed.
我们给出了任意阶的Reed-Muller码的最小距离公式的另一种证明。它导致了一个完整的表征最小权码字的投影里德-穆勒码。然后用它来确定投影里德-穆勒码的最小权码字数。讨论了射影里德-穆勒码维数的各种公式及其等价性。
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引用次数: 1
The MacWilliams identity for the skew rank metric 倾斜等级度量的MacWilliams恒等式
4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-01-01 DOI: 10.3934/amc.2023045
Izzy Friedlander, Thanasis Bouganis, Maximilien Gadouleau
The weight distribution of an error correcting code is a crucial statistic in determining its performance. One key tool for relating the weight of a code to that of it's dual is the MacWilliams Identity, first developed for the Hamming metric. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via (generalised) Krawtchouk polynomials. The functional transformation form can in particular be used to derive important moment identities for the weight distribution of codes. In this paper, we focus on codes in the skew rank metric. In these codes, the codewords are skew-symmetric matrices, and the distance between two matrices is the skew rank metric, which is half the rank of their difference. This paper develops a $ q $-analog MacWilliams Identity in the form of a functional transformation for codes based on skew-symmetric matrices under their associated skew rank metric. The method introduces a skew-$ q $ algebra and uses generalised Krawtchouk polynomials. Based on this new MacWilliams Identity, we then derive several moments of the skew rank distribution for these codes.
纠错码的权值分布是决定纠错码性能的重要统计量。将码的权重与其对偶的权重联系起来的一个关键工具是MacWilliams恒等式,它最初是为汉明度量开发的。这个恒等式有两种形式:一种是权重枚举数的函数变换,而另一种是通过(广义)克劳楚克多项式的权重分布的直接关系。该函数变换形式可用于推导码权分布的重要矩恒等式。在本文中,我们主要研究歪斜秩度量中的代码。在这些码中,码字是偏对称矩阵,两个矩阵之间的距离是偏秩度量,它是它们差的秩的一半。本文给出了基于斜对称矩阵的码在其相关的斜秩度量下的函数变换形式的$ q $-模拟MacWilliams恒等式。该方法引入了一个偏q代数,并使用了广义克劳楚克多项式。基于这个新的MacWilliams恒等式,我们得到了这些码的偏秩分布的几个矩。
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引用次数: 0
Polynomial representation of additive cyclic codes and new quantum codes 加性循环码与新量子码的多项式表示
4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-01-01 DOI: 10.3934/amc.2023036
Reza Dastbasteh, Khalil Shivji
We give a polynomial representation for additive cyclic codes over $ mathbb{F}_{p^2} $. This representation will be applied to uniquely present each additive cyclic code by at most two generator polynomials. We determine the generator polynomials of all different additive cyclic codes. A minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic codes over $ mathbb{F}_p $. We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal additive cyclic codes over $ mathbb{F}_{p^2} $. Finally, we present ten record-breaking binary quantum codes after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive cyclic codes over $ mathbb{F}_{4} $.
给出了$ mathbb{F}_{p^2} $上的加性循环码的多项式表示。这种表示将被应用于通过最多两个生成器多项式唯一地表示每个加性循环码。我们确定了所有不同加性循环码的生成多项式。使用$ mathbb{F}_p $上的线性循环码也提供了加性循环码的最小距离下界。我们对$ mathbb{F}_{p^2} $上的所有辛自对偶、自正交和近自正交加性循环码进行了分类。最后,我们将量子结构应用于$ mathbb{F}_{4} $上的自正交和近自正交加性循环码,得到了10个破纪录的二进制量子码。
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引用次数: 0
Construction of extremal $ mathbb{Z}_{4} $-codes using a neighborhood search algorithm 用邻域搜索算法构造极值$ mathbb{Z}_{4} $-码
4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-01-01 DOI: 10.3934/amc.2023039
Dean Crnković, Matteo Mravić, Sanja Rukavina
In this paper, we present a method for constructing extremal $ mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type Ⅱ $ mathbb{Z}_{4} $-codes of lengths 32 and 40. For the length 32, at least 182 new Type Ⅱ extremal $ mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ kinleft{9,10,12,13,14,15,16right} $ are constructed. In addition, we obtained at least 762 new extremal Type Ⅰ $ mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ kinleft{7,9,10,12,13,14,15,16right} $. For the length 40, constructed extremal $ mathbb{Z}_{4} $-codes are of types $ 4^{k}2^{40-2k} $, $ kinleft{7,10,11,15,16right} $. There are at least 40 new Type Ⅱ extremal $ mathbb{Z}_{4} $-codes, and at least 4144 new Type Ⅰ extremal $ mathbb{Z}_{4} $-codes.
本文提出了一种基于随机邻域搜索构造极值$ mathbb{Z}_{4} $-码的方法。此方法用于查找新的极值类型Ⅰ和类型Ⅱ$ mathbb{Z}_{4} $-长度为32和40的代码。对于长度32,至少构造182个新的类型Ⅱ极值$ mathbb{Z}_{4} $-类型$ 4^{k}2^{32-2k} $, $ kin左{9,10,12,13,14,15,16右}$的代码。此外,我们得到了至少762个新的极值类型Ⅰ$ mathbb{Z}_{4} $-类型为$ 4^{k}2^{32-2k} $, $ k In 左{7,9,10,12,13,14,15,16右}$的代码。对于长度40,构造的极值$ mathbb{Z}_{4} $-代码的类型为$ 4^{k}2^{40-2k} $, $ kin左{7,10,11,15,16右}$。至少有40个新的TypeⅡextremal $ mathbb{Z}_{4} $-代码,以及至少4144个新的TypeⅠextremal $ mathbb{Z}_{4} $-代码。
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引用次数: 0
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Advances in Mathematics of Communications
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