[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84419-en":3,"doc-seo-84419-105":29,"detail-sidebar-cat-0-en-105":95},{"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},84419,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Quantum Codes from Group Codes","Study presents a unified construction of linear codes and quantum error-correcting codes (QECCs) from group rings over finite fields. Leveraging the algebraic structure of group rings, it covers cyclic, dihedral, direct-product, and semidirect-product groups within one framework. Necessary and sufficient self-orthogonality criteria are derived under Euclidean, Hermitian, and symplectic inner products, and inequivalent codes from non-isomorphic groups of the same order are demonstrated. Explicit quantum code constructions, block-matrix generating forms, and an infinite Kronecker-product family are provided.","arXiv :2412 . 136 16v 3 [ cs .IT] 11 Jul 2026  \nQuantum Codes from Group Codes  \nTushar Bag 1 ,2 , Daniel Panario3  \n1. Department of Mathematics, SRM University-AP, Amaravati 522240, Andhra Pradesh, India  \n2. Inria, ENS de Lyon, Lyon 1, LIP, 69342, Lyon cedex 07, France  \n3. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada.  \nJuly 14, 2026  \nAbstract  \nWe study linear codes and quantum error-correcting codes (QECCs) constructed from group rings over finite fields. Using the algebraic structure of group rings, we give a single framework for codes over several group structures, including cyclic, dihedral, direct-product, and semidirect-product groups. We establish necessary and sufficient conditions for these group codes to be self-orthogonal under the Euclidean, Hermitian, and symplectic inner products. We show that non-isomorphic groups of the same order can generate inequivalent codes with distinct parameters, and we support this with explicit computational comparisons. Using these structural results, we give explicit constructions of quantum codes and provide new examples that match or improve upon the best known parameters. In particular, we describe explicit block-matrix forms of the generating matrices for dihedral and direct-product groups, and we use a Kronecker-product construction to obtain an infinite family of self-orthogonal group codes together with the corresponding QECCs.  \nKeywords: Group rings, quantum error-correcting codes (QECCs) .  \nMathematics Subject Classification (2010): 94B05, 94B15, 94B60 .  \n1 Introduction  \nGroup codes were introduced by Berman [3, 4], with an initial focus on abelian and semisimple abelian codes. Significant theoretical advances were established in subsequent works [5, 19, 20, 21, 28, 29, 35] . In particular, Hurley [22] constructed an explicit isomorphism between a group ring and a matrix ring, which simplifies the encoding and analysis of group codes. This matrix representation is the main algebraic tool used in our study. More recently, group codes employing this matrix approach have been explored by Dougherty et al. [13, 14, 15] and by Yu and Zhu [37, 38] . Borello and Willems [7] proved that group codes over finite fields are asymptotically good in every characteristic. Borello’s lecture notes [8] provide an overview of the algebraic structure of group codes and their connections with classical linear codes. Related work by Borello and Jamous [6] examines the automorphism groups of binary linear codes, while group codes have also proven useful for constructing good linear codes [23, 26] .  \nQuantum error-correcting codes (QECCs) protect quantum information against decoherence and quantum noise. Quantum computers offer significant computational potential, but they remain susceptible to errors caused by environmental interactions, and QECCs are a central tool for protecting quantum information in both computing and communication. The foundational concepts of QECCs were introduced by Shor, Steane, and Calderbank [12, 33, 34], and the Calderbank–Shor–Steane (CSS) framework [11] continues to play a fundamental role in  \nEmail: [tusharbag2011@gmail.com](tusharbag2011@gmail.com) (T. Bag) [corresponding author], daniel@math.carleton.ca (D. Panario)  \nquantum coding theory. Techniques for constructing non-binary quantum codes from classical self-orthogonal codes have been developed in subsequent works [1, 2, 27] .  \nOur motivation for studying linear codes through group rings, and for using them to construct quantum codes, is the following. For a fixed length, group rings provide several different generating matrices for a code. For example, to construct a code of length 12 over F q, one can use the group ring Fq[G] for any group G of order 12 . Up to isomorphism there are five such groups: the cyclic group C12 , the dihedral group D6 , the direct product C2 ×C6 , the alternating group A4 , and the dicyclic group C3 ⋊ C4 . Here Cn is the cyc","cbCaia4ryUvRElOW","https://ap.wps.com/l/cbCaia4ryUvRElOW","pdf",392490,1,28,"English","en",105,"# Introduction\n# Preliminaries\n## Group rings and linear codes\n## Quantum error-correcting codes (QECCs)\n# Group code constructions\n## Block-matrix generation from group structures\n## Ordering effects on generating matrices\n# Inequivalent codes from non-isomorphic groups\n# Self-orthogonality criteria and quantum constructions\n## Euclidean, Hermitian, and symplectic inner products\n# Conclusion","[{\"question\":\"How does the paper build quantum error-correcting codes from group codes?\",\"answer\":\"It constructs classical linear codes from group rings using Hurley’s matrix homomorphism, then uses self-orthogonality conditions under Euclidean, Hermitian, or symplectic inner products to derive corresponding QECCs.\"},{\"question\":\"What groups are covered by the unified group-ring framework?\",\"answer\":\"The framework supports multiple group structures, including cyclic, dihedral, direct-product, and semidirect-product groups, enabling code design across distinct non-isomorphic group families.\"},{\"question\":\"Why can non-isomorphic groups of the same order yield inequivalent codes?\",\"answer\":\"The paper establishes that group structure influences the resulting generating matrices and parameters; computational comparisons show that different non-isomorphic groups with the same order can produce inequivalent codes.\"},{\"question\":\"What construction methods are used to obtain explicit families of codes?\",\"answer\":\"It provides explicit block-matrix forms for generating matrices in dihedral and direct-product cases and uses a Kronecker-product construction to obtain an infinite family of self-orthogonal group codes together with their QECCs.\"}]",1784195505,71,{"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":90,"head_meta":92,"extra_data":94,"updated_unix":27},"quantum-codes-from-group-codes","",{"@graph":35,"@context":89},[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/quantum-codes-from-group-codes/84419/",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",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"How does the paper build quantum error-correcting codes from group codes?","Question",{"text":75,"@type":76},"It constructs classical linear codes from group rings using Hurley’s matrix homomorphism, then uses self-orthogonality conditions under Euclidean, Hermitian, or symplectic inner products to derive corresponding QECCs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What groups are covered by the unified group-ring framework?",{"text":80,"@type":76},"The framework supports multiple group structures, including cyclic, dihedral, direct-product, and semidirect-product groups, enabling code design across distinct non-isomorphic group families.",{"name":82,"@type":73,"acceptedAnswer":83},"Why can non-isomorphic groups of the same order yield inequivalent codes?",{"text":84,"@type":76},"The paper establishes that group structure influences the resulting generating matrices and parameters; computational comparisons show that different non-isomorphic groups with the same order can produce inequivalent codes.",{"name":86,"@type":73,"acceptedAnswer":87},"What construction methods are used to obtain explicit families of codes?",{"text":88,"@type":76},"It provides explicit block-matrix forms for generating matrices in dihedral and direct-product cases and uses a Kronecker-product construction to obtain an infinite family of self-orthogonal group codes together with their QECCs.","https://schema.org",{"og:url":51,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},"Exam",70,"exam",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},5,"Comic",60,"comic",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},6,"Technology",50,"technology",{"id":120,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":122,"slug":123},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":125,"slug":126},30,"research-report",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":130,"slug":131},9,"Religion & Spirituality",20,"religion-spirituality",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":130,"slug":134},"World Cup","world-cup",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":136,"slug":138},10,"Lifestyle","lifestyle",{"id":140,"doc_module":4,"doc_module_name":45,"category_name":141,"show_sort_weight":110,"slug":142},19,"General","general"]