Constacyclic codes over $${{\mathbb {Z}}_2[u]}/{\langle u^2\rangle }\times {{\mathbb {Z}}_2[u]}/{\langle u^3\rangle }$$ and the MacWilliams identities

IF 0.6 4区 工程技术 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS Applicable Algebra in Engineering Communication and Computing Pub Date : 2024-07-04 DOI:10.1007/s00200-024-00662-6
Vidya Sagar, Ankit Yadav, Ritumoni Sarma
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Abstract

In this article, we deal with additive codes over the Frobenius ring \({\mathcal {R}}_{2}{\mathcal {R}}_{3}:=\frac{{\mathbb {Z}}_{2}[u]}{\langle u^2 \rangle }\times \frac{{\mathbb {Z}}_{2}[u]}{\langle u^3 \rangle }\). First, we study constacyclic codes over \({\mathcal {R}}_2\) and \({\mathcal {R}}_3\) and find their generator polynomials. With the help of these generator polynomials, we determine the structure of constacyclic codes over \({\mathcal {R}}_2{\mathcal {R}}_3\). We use Gray maps to show that constacyclic codes over \({\mathcal {R}}_{2}{\mathcal {R}}_{3}\) are essentially binary generalized quasi-cyclic codes. Moreover, we obtain a number of binary codes with good parameters from these \({\mathcal {R}}_{2}{\mathcal {R}}_{3}\)-constacyclic codes. Besides, several weight enumerators are computed, and the corresponding MacWilliams identities are established.

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关于 $${{{mathbb {Z}}_2[u]}/{{langle u^2\rangle }/{{{mathbb {Z}}_2[u]}/{{langle u^3\rangle }$$ 和麦克威廉斯等式的恒环码
在本文中,我们将讨论弗罗贝尼斯环({\mathcal {R}_{2}{mathcal {R}_{3}:=\frac{{\mathbb {Z}_{2}[u]}{langle u^2 \rangle }/times)上的加法码。首先,我们研究在 \({\mathcal {R}}_2\) 和 \({\mathcal {R}}_3\) 上的constacyclic编码,并找到它们的生成器多项式。在这些生成器多项式的帮助下,我们确定了 \({\mathcal {R}}_2{\mathcal {R}}_3\) 上的 Constacyclic 码的结构。我们使用格雷映射来证明在 \({\mathcal {R}_{2}{\mathcal {R}_{3}\) 上的constacyclic码本质上是二进制广义准循环码。而且,我们从这些 \({\mathcal {R}_{2}{mathcal {R}_{3}}/)-constacyclic码中得到了许多具有良好参数的二进制码。此外,我们还计算了几种权值枚举器,并建立了相应的麦克威廉斯(MacWilliams)等价性。
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来源期刊
Applicable Algebra in Engineering Communication and Computing
Applicable Algebra in Engineering Communication and Computing 工程技术-计算机:跨学科应用
CiteScore
2.90
自引率
14.30%
发文量
48
审稿时长
>12 weeks
期刊介绍: Algebra is a common language for many scientific domains. In developing this language mathematicians prove theorems and design methods which demonstrate the applicability of algebra. Using this language scientists in many fields find algebra indispensable to create methods, techniques and tools to solve their specific problems. Applicable Algebra in Engineering, Communication and Computing will publish mathematically rigorous, original research papers reporting on algebraic methods and techniques relevant to all domains concerned with computers, intelligent systems and communications. Its scope includes, but is not limited to, vision, robotics, system design, fault tolerance and dependability of systems, VLSI technology, signal processing, signal theory, coding, error control techniques, cryptography, protocol specification, networks, software engineering, arithmetics, algorithms, complexity, computer algebra, programming languages, logic and functional programming, algebraic specification, term rewriting systems, theorem proving, graphics, modeling, knowledge engineering, expert systems, and artificial intelligence methodology. Purely theoretical papers will not primarily be sought, but papers dealing with problems in such domains as commutative or non-commutative algebra, group theory, field theory, or real algebraic geometry, which are of interest for applications in the above mentioned fields are relevant for this journal. On the practical side, technology and know-how transfer papers from engineering which either stimulate or illustrate research in applicable algebra are within the scope of the journal.
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