[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82201-en":3,"doc-seo-82201-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82201,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Two-dimensional Constacyclic Codes Over Finite Chain Rings","Two-dimensional (λ,µ)-constacyclic codes of length ℓm over finite chain rings are studied, assuming residue field Fq with q ≡ 1 (mod rm), where r is the multiplicative order of µ̄. The algebraic structure of these codes is derived by analyzing primitive idempotents in the finite chain ring to obtain code generators. A further result gives a condition ensuring that two-dimensional constacyclic codes achieve maximum hamming distance with respect to rank (MHDR) over finite chain rings.","arXiv :2607 .09 1 17v 1 [ cs .IT] 10 Jul 2026  \nTwo-dimensional constacyclic codes over finite chain  \nrings  \nVaishali Singh 1 , Sucheta Dutt2 and Ridhima Thakral3 ,∗  \n[vaishali.phd24maths@pec.edu.in](vaishali.phd24maths@pec.edu.in)[ ](vaishali.phd24maths@pec.edu.in)[sucheta@pec.edu.in](sucheta@pec.edu.in)  \n[ridhimathakral@pec.edu.in](ridhimathakral@pec.edu.in)  \n1 ,2 ,3 Department of Mathematics,  \nPunjab Engineering College (Deemed to be University), Sector 12, Chandigarh 160012, India  \nAbstract  \nThe main focus of this paper is on the algebraic structure of two-dimensional (λ,µ)-constacyclic codes of length ℓm over finite chain rings with residue field Fq , where q ≡ 1 (mod rm) and r denotes the multiplicative order of µ¯ . In this paper, the structure of two-dimensional (λ,µ)-constacyclic codes is obtained. Our approach relies on analysing primitive idempotents within the finite chain ring to determine the generators of these codes. We also find the condition under which two-dimensional constacyclic codes are maximum hamming distance with respect to rank (MHDR) over finite chain rings.  \n1 Introduction  \nThe practical need for reliable data transmission and storage in digital systems has motivated a deep and extensive study of error-correcting codes. The algebraic framework of cyclic and constacyclic codes has made them one of the most extensively studied classes of linear error-correcting codes. Their structural properties enable efficient encoding and decoding procedures, making them suitable for a wide range of communication and storage applications. Therefore, extensive research has been conducted on the structural properties and generator characterisations of these codes over fields and rings. The exploration of codes over finite chain rings has gained prominence as finite chain rings form a superset of finite fields. Later, two-dimensional cyclic and constacyclic codes developed as a natural generalisation of classical cyclic codes over finite fields as well as finite chain rings.  \nIn 1977, Imai [4] was the first to propose the idea of two-dimensional binary cyclic codes and has since attracted significant attention due to its theoretical richness and practical  \nrelevance in multi-dimensional signal processing and data transmission. In 2012, C.Güneri 2020 Mathematics Subject Classication. Primary: 11T71; Secondary: 94B05 .  \nKeywords: Primitive idempotents, finite chain rings, two-dimensional constacyclic codes, MHDR codes.  \n*Corresponding author: Ridhima Thakral.  \nand F. Özbudak [3] showed that two-dimensional cyclic codes are algebraically equivalent to a special class of quasi-cyclic codes. In 2016, a study focused on two-dimensional cyclic codes of length s.2k and their dual codes over a finite field was carried out by Z. Sepasdarand K. Khashyarmanesh [8] . In 2017, Z. Sepasdar [7] also derived the generator matrix corresponding to two-dimensional cyclic codes of any given length over finite fields. An algebraic structure of some two-dimensional constacyclic codes of length 2ps.2k was given by Z. Rajabi and K. Khashyarmanesh [6] in 2017 . In 2021, S.Bhardwaj and M.Raka [1] analyzed (α,β)-constacyclic codes of length sl over Fq . In 2023, D.Garg and S.Dutt [2] obtained the generators of two-dimensional cyclic codes over finite chain rings using primitive idempotents.  \nMotivated by these developments, the present work focuses on the construction of two-dimensional (λ,µ)-constacyclic codes of length ℓm over finite chain rings with residue field Fq , where q ≡ 1 (mod rm) and r denotes the multiplicative order of µ¯ .  \nThe remainder of this paper is structured as follows. Necessary definitions and foundational results related to constacyclic codes over finite chain rings are given in section 2 . Generators of two-dimensional (λ,µ)-constacyclic codes of length ℓm over finite chain rings with the help of primitive idempotents are obtained in section 3 . A condition for two-dimensional constacyclic codes to be ","cbCaidC1llUkbX2Y","https://ap.wps.com/l/cbCaidC1llUkbX2Y","pdf",401285,1,11,"English","en",105,"# Introduction\n## Background and Motivation\n# Preliminaries\n## Constacyclic Codes over Finite Chain Rings\n## Idempotents and Primitive Idempotents\n## Finite Chain Rings and Residue Fields\n## Torsion Codes\n# Generators of Two-dimensional (λ,µ)-Constacyclic Codes\n# MHDR Condition for Two-dimensional Constacyclic Codes\n# Conclusion","[{\"question\":\"What problem does the paper address about two-dimensional constacyclic codes?\",\"answer\":\"It identifies a condition under which these two-dimensional constacyclic codes are maximum hamming distance with respect to rank (MHDR) over finite chain rings.\"},{\"question\":\"How does the paper construct generators of the codes?\",\"answer\":\"Generators are derived by analyzing primitive idempotents inside the finite chain ring, which leads to explicit generator descriptions for the (λ,µ)-constacyclic codes.\"},{\"question\":\"What assumptions are imposed on the parameters and the residue field Fq?\",\"answer\":\"The study considers residue field Fq satisfying q ≡ 1 (mod rm), where r is the multiplicative order of µ̄, and the codes have length ℓm over the finite chain ring.\"}]",1784178791,28,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"two-dimensional-constacyclic-codes-over-finite-chain-rings","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/two-dimensional-constacyclic-codes-over-finite-chain-rings/82201/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address about two-dimensional constacyclic codes?","Question",{"text":75,"@type":76},"It identifies a condition under which these two-dimensional constacyclic codes are maximum hamming distance with respect to rank (MHDR) over finite chain rings.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper construct generators of the codes?",{"text":80,"@type":76},"Generators are derived by analyzing primitive idempotents inside the finite chain ring, which leads to explicit generator descriptions for the (λ,µ)-constacyclic codes.",{"name":82,"@type":73,"acceptedAnswer":83},"What assumptions are imposed on the parameters and the residue field Fq?",{"text":84,"@type":76},"The study considers residue field Fq satisfying q ≡ 1 (mod rm), where r is the multiplicative order of µ̄, and the codes have length ℓm over the finite chain 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