[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82639-en":3,"doc-seo-82639-105":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},82639,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","On the structure of constacyclic codes over finite chain rings","Explicit generators are constructed for a λ-constacyclic code C of arbitrary length over a finite chain ring R by using minimum degree polynomials in the quotient ring R[x]/⟨x^ℓ−λ⟩. The construction is shown to achieve the minimum possible number of generators and its generator properties are used to derive a minimal spanning set. The code rank equals ℓ−n0, where n0 is the relevant minimum degree polynomial. Necessary and sufficient MHDR and MDS conditions are characterized via a torsion code over the residue field Fq.","arXiv :2607 .0 177 1v 1 [ cs .IT] 2 Jul 2026  \nOn the structure of constacyclic codes over finite chain  \nrings  \nVaishali Singh\\# 1 , Sucheta Dutt\\# 2 and Ridhima Thakral\\# ∗ ,3  \n1 ,2 ,3 Department of Mathematics,  \nPunjab Engineering College (Deemed to be University), Sector 12, Chandigarh 160012, India  \nAbstract  \nIn the present paper, we provide an explicit construction for generators of a λ-constacyclic code C of arbitrary length ℓ over a finite chain ring(FCR) R in terms of certain minimum degree polynomials of the ring R [x]/⟨xℓ − λ⟩ . Moreover, the proposed construction achieves the minimum possible number of generators. We prove certain properties of this set of generators, using which we obtain a minimal spanning set of C. We also obtain that the rank of C is ℓ − n0 , where n0 is the degree of the minimal degree polynomial in C. Finally, we derive necessary and sufficient conditions under which an arbitrary length λ-constacyclic code C over R is Maximum Hamming Distance with respect to Rank(MHDR) as well as Maximum Distance Separable(MDS) in terms of a torsion code of C over the residue field Fq of  \nR. We further determine the exact values for n0 for which C over R is MHDR.  \n1 Introduction  \nReliable transmission of information across noisy channels has long demanded mathematical tools and error-correcting codes occupy a central place among them. Within this domain, the class of cyclic and constacyclic codes received considerable attention due to their close connection between the codewords and ideals of the polynomial ring.  \nEarly studies on cyclic and negacyclic codes over finite fields were established in the 1960s by Berlekamp [19] . Subsequently, in 1991, Castagnoli et al. [4] and Lint [18] investigated repeated root cyclic codes. In 1994, it was observed by Hammons et al. [1] that certain nonlinear binary codes derived from linear codes over Z4 using Gray maps. An advancement in coding theory was achieved with the introduction of finite chain rings(FCR) as an underlying algebraic structure. In this direction, the theory of linear and cyclic codes over FCR was developed by Norton and Sălăgean [23, 22] in 2000 . They also found the Hamming distances of such codes. In 2004, Dinh and López-Permouth [17] studied the structural properties of cyclic and negacyclic codes over a FCR, particularly for those lengths that do not divide the characteristic of the residue field of the FCR. Further  \n2020 Mathematics Subject Classication. Primary: 11T71,94B15,94B65; Secondary: 94B05 . Keywords: Finite chain rings, constacyclic codes, MHDR codes, MDS codes.  \n*Corresponding author: Ridhima Thakral.  \nstudies in 2005 by Dinh [9] focused on negacyclic codes defined over Galois rings with length 2s. In 2006, it was established by Sălăgean [25] that repeated root cyclic and negacyclic codes defined over FCR are generally not principally generated. In 2008, Dinh studied the structural properties of prime power length negacyclic codes over finite fields [10] . In 2009, Dinh [11] investigated length 2s constacyclic codes over Galois extension rings of F2 + uF2 . Continuing this line of research, Dinh [12] in 2010, analyzed the structural characterisation and Hamming distances of (α+uβ)-constacyclic codes of length ps over the ring Fpm+uFpm . In 2012, Dinh[13] further explored repeated-root constacyclic codes by examining the length 2ps over finite fields. During the same year, Chen et al. [7] provided an explicit characterisation of generator polynomials associated with constacyclic codes of length ltps over a finite field, where p denotes the field characteristic and l is a prime different from p. In 2013, Cao [2] examined arbitrary length (1 + wγ)-constacyclic codes over FCR. Also in that year, Dinh [14] investigated length 3ps repeated-root constacyclic codes over finite fields, focusing on their structural and dual properties. Further developments were made in 2014 by Chen et al. [5], who characterised repeated-root constacyc","cbCaiqy2JWEXaMoy","https://ap.wps.com/l/cbCaiqy2JWEXaMoy","pdf",473809,1,13,"English","en",105,"# Introduction\n## Background and related work\n# Main construction and generator sets\n## Minimum degree polynomial framework\n## Minimal spanning sets and rank\n# MHDR and MDS characterizations","[{\"question\":\"How are generators of a λ-constacyclic code constructed over a finite chain ring?\",\"answer\":\"Generators of a λ-constacyclic code of arbitrary length are explicitly constructed using certain minimum degree polynomials in the ring R[x]/⟨x^ℓ−λ⟩.\"},{\"question\":\"What is the minimum number of generators achieved by the proposed construction?\",\"answer\":\"The proposed generator construction attains the minimum possible number of generators for the code C.\"},{\"question\":\"How are MHDR and MDS conditions determined for λ-constacyclic codes over R?\",\"answer\":\"Necessary and sufficient conditions are derived in terms of a torsion code of C over the residue field Fq of R, and exact values of the key degree parameter n0 for the MHDR property are obtained.\"}]",1784181973,33,{"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},"on-the-structure-of-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/on-the-structure-of-constacyclic-codes-over-finite-chain-rings/82639/",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},"How are generators of a λ-constacyclic code constructed over a finite chain ring?","Question",{"text":75,"@type":76},"Generators of a λ-constacyclic code of arbitrary length are explicitly constructed using certain minimum degree polynomials in the ring R[x]/⟨x^ℓ−λ⟩.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the minimum number of generators achieved by the proposed construction?",{"text":80,"@type":76},"The proposed generator construction attains the minimum possible number of generators for the code C.",{"name":82,"@type":73,"acceptedAnswer":83},"How are MHDR and MDS conditions determined for λ-constacyclic codes over R?",{"text":84,"@type":76},"Necessary and sufficient conditions are derived in terms of a torsion code of C over the residue field Fq of R, and exact values of the key degree parameter n0 for the MHDR property are 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