Hai Q. Dinh, Mohammad Ashraf, Washiqur Rehman, Ghulam Mohammad, Mohd Asim
{"title":"On reversible DNA codes over the ring $${\\mathbb {Z}}_4[u,v]/\\langle u^2-2,uv-2,v^2,2u,2v\\rangle$$ based on the deletion distance","authors":"Hai Q. Dinh, Mohammad Ashraf, Washiqur Rehman, Ghulam Mohammad, Mohd Asim","doi":"10.1007/s00200-024-00661-7","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\({\\mathfrak {R}}= {\\mathbb {Z}}_4[u,v]/\\langle u^2-2,uv-2,v^2,2u,2v\\rangle\\)</span> be a ring, where <span>\\({\\mathbb {Z}}_{4}\\)</span> is a ring of integers modulo 4. This ring <span>\\({\\mathfrak {R}}\\)</span> is a local non-chain ring of characteristic 4. The main objective of this article is to construct reversible cyclic codes of odd length <i>n</i> over the ring <span>\\({\\mathfrak {R}}.\\)</span> Employing these reversible cyclic codes, we obtain reversible cyclic DNA codes of length <i>n</i>, based on the deletion distance over the ring <span>\\({\\mathfrak {R}}.\\)</span> We also construct a bijection <span>\\(\\Gamma\\)</span> between the elements of the ring <span>\\({\\mathfrak {R}}\\)</span> and <span>\\(S_{D_{16}}.\\)</span> As an application of <span>\\(\\Gamma ,\\)</span> the reversibility problem which occurs in DNA <i>k</i>-bases has been solved. Moreover, we introduce a Gray map <span>\\(\\Psi _{\\hom }:{\\mathfrak {R}}^{n}\\rightarrow {\\mathbb {F}}_{2}^{8n}\\)</span> with respect to homogeneous weight <span>\\(w_{\\hom }\\)</span> over the ring <span>\\({\\mathfrak {R}}\\)</span>. Further, we discuss the <i>GC</i>-content of DNA cyclic codes and their deletion distance. Moreover, we provide some examples of reversible DNA cyclic codes.</p>","PeriodicalId":50742,"journal":{"name":"Applicable Algebra in Engineering Communication and Computing","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-07-09","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/s00200-024-00661-7","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
Let \({\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle\) be a ring, where \({\mathbb {Z}}_{4}\) is a ring of integers modulo 4. This ring \({\mathfrak {R}}\) is a local non-chain ring of characteristic 4. The main objective of this article is to construct reversible cyclic codes of odd length n over the ring \({\mathfrak {R}}.\) Employing these reversible cyclic codes, we obtain reversible cyclic DNA codes of length n, based on the deletion distance over the ring \({\mathfrak {R}}.\) We also construct a bijection \(\Gamma\) between the elements of the ring \({\mathfrak {R}}\) and \(S_{D_{16}}.\) As an application of \(\Gamma ,\) the reversibility problem which occurs in DNA k-bases has been solved. Moreover, we introduce a Gray map \(\Psi _{\hom }:{\mathfrak {R}}^{n}\rightarrow {\mathbb {F}}_{2}^{8n}\) with respect to homogeneous weight \(w_{\hom }\) over the ring \({\mathfrak {R}}\). Further, we discuss the GC-content of DNA cyclic codes and their deletion distance. Moreover, we provide some examples of reversible DNA cyclic codes.
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.