{"title":"Constacyclic codes over $${{\\mathbb {Z}}_2[u]}/{\\langle u^2\\rangle }\\times {{\\mathbb {Z}}_2[u]}/{\\langle u^3\\rangle }$$ and the MacWilliams identities","authors":"Vidya Sagar, Ankit Yadav, Ritumoni Sarma","doi":"10.1007/s00200-024-00662-6","DOIUrl":null,"url":null,"abstract":"<p>In this article, we deal with additive codes over the Frobenius ring <span>\\({\\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 }\\)</span>. First, we study constacyclic codes over <span>\\({\\mathcal {R}}_2\\)</span> and <span>\\({\\mathcal {R}}_3\\)</span> and find their generator polynomials. With the help of these generator polynomials, we determine the structure of constacyclic codes over <span>\\({\\mathcal {R}}_2{\\mathcal {R}}_3\\)</span>. We use Gray maps to show that constacyclic codes over <span>\\({\\mathcal {R}}_{2}{\\mathcal {R}}_{3}\\)</span> are essentially binary generalized quasi-cyclic codes. Moreover, we obtain a number of binary codes with good parameters from these <span>\\({\\mathcal {R}}_{2}{\\mathcal {R}}_{3}\\)</span>-constacyclic codes. Besides, several weight enumerators are computed, and the corresponding MacWilliams identities are established.</p>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":"44 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applicable Algebra in Engineering Communication and Computing","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1007/s00200-024-00662-6","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
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.
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.