[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83655-en":3,"doc-seo-83655-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83655,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Generalized Rank Weight and Extended Generalized Poset Weight for Codes Over Rings: A Galois Connection Approach","Study generalized rank weights (GRWs) and extended generalized poset weights (EGPWs) for codes over rings using a Galois connection framework. Reformulate coding-theoretic properties—wire-tap type II security drops, Gabidulin–Delsarte relationships, Singleton and MDS-type bounds, MRD/near MRD/i-MRD/MRD-dual characterizations, and evasive subspace behavior—via Galois connections. Develop GRWs and rank profiles for modules over principal ideal rings and chain rings, proving singleton bounds, Wei-type duality, and scattered bounds. Introduce EGPWs/extended poset profiles for modules with composition series and prove Wei-type duality over quasi-Frobenius rings.","arXiv :2607 .02377v 1 [ cs .IT] 2 Jul 2026  \nGeneralized Rank Weight and Extended Generalized Poset Weight Defined For Codes Over Rings: A Galois Connection Approach  \nYang Xu 1 Haibin Kan2 Guangyue Han3  \nJuly 3, 2026  \nAbstract—In this paper, we study generalized rank weights (GRWs) and extended generalized poset weight (EGPWs) of codes over rings via a Galois connection approach. First, we show that various coding-theoretic properties related to generalized weights, including security drops of a code employed in wire-tap channel of type II, connections between generalized weights of a Gabidulin code and its associated Delsarte code, (generalized) Singleton bound, MDS discrepancy of a code, characterizations of MDS, near MDS, i-MDS, MRD, near MRD, i-MRD,(dually) quasi-MRD codes as well as evasive property of subspaces, can be reformulated in terms of Galois connections. Next, we study GRWs and rank profiles defined for modules over principal ideal rings, especially those over chain rings. Generalizing GRWs defined for vector spaces over fields, we establish a singleton bound and a Wei-type duality theorem, characterize MRD, near MRD and dually quasi-MRD codes and determine their GRWs; moreover, we characterize i-MRD codes and establish a scattered bound for (h, h)-evasive codes over chain rings, generalizing counterpart result established for vector space over finite fields. Finally, we propose and study EGPWs and extended poset profiles defined for modules with a composition series, which in fact form a Galois connection. Generalizing EGPWs defined for modules over finite Galois rings, we establish a Wei-type duality theorem for modules over arbitrary quasi-Frobenius rings, which unifies the two Wei-type duality theorems derived in both [32] and [33] .  \n1 Introduction  \nIn 1991, motivated by applications from information-theoretic security, Wei proposed and studied generalized Hamming weights (GHWs) of linear codes. Roughly speaking, for a linear code C , the r-th GHW of C is defined as the minimal Hamming weight of all the r-dimensional subcodes of C. It has been shown in [34] that GHWs characterize both the performance of a code on the wiretap channel of type II and the performance of the code as a t-resilient function. GHWs have been widely used to gauge the security performances of linear codes for secret sharing, secure network coding or distributed data storage, see, e.g., [5, 15, 24] and references therein.  \n1 Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai 200433, China.  \nShanghai Engineering Research Center of Blockchain, Shanghai 200433, [China. E-mail:xuyyang@fudan.edu.cn](China. E-mail:xuyyang@fudan.edu.cn)  \n2 Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai 200433, China.  \nShanghai Engineering Research Center of Blockchain, Shanghai 200433, China.  \nYiwu Research Institute of Fudan University, Yiwu City, Zhejiang 322000, [China. E-mail:hbkan@fudan.edu.cn](China. E-mail:hbkan@fudan.edu.cn)  \n3 Department of Mathematics, Faculty of Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China. E-mail:ghan@hku.hk  \nTheoretically, GHWs have been used to characterize various families of codes such as r-th rank MDS codes (r-MDS codes for short), AsMDS codes, and dually AMDS codes (or equivalently, near MDS codes) (see [14, 34]) . Some important algebraic and combinatorial properties of GHWs, such as monotonicity and Wei’s duality theorem, provide powerful tools for studying these codes (see [14, 34] for more details) . Recently in [32], Tang proposed and studied an extension of generalized Hamming weights (EGHWs) for codes over Zpm . In [32], among others, the author establishes two Wei-type duality theorems and several bounds for Zpm-linear codes.  \nExtending the notion of GHWs, generalized weights with respect to rank metric and poset metric have become topics of gr","cbCaigB0LuqSRcMx","https://ap.wps.com/l/cbCaigB0LuqSRcMx","pdf",428565,4,1,33,"English","en",105,"# Introduction\n## Generalized Hamming weights and their security meaning\n## Rank-metric generalized weights and applications\n## Extensions to codes over rings\n## Poset-metric motivation","[{\"question\":\"What problem does the paper address regarding generalized weights of codes?\",\"answer\":\"The paper investigates generalized rank weights and extended generalized poset weights for codes over rings, aiming to unify and reinterpret multiple coding-theoretic properties using a Galois connection approach.\"},{\"question\":\"How does the Galois connection framework relate to coding-theoretic properties?\",\"answer\":\"It reformulates diverse properties—such as Singleton/MDS-type bounds and classifications of MRD and near-MRD families, as well as evasive subspace behavior—into statements expressed through Galois connections.\"},{\"question\":\"Which algebraic structures are considered when extending GRWs and EGPWs?\",\"answer\":\"GRWs and rank profiles are developed for modules over principal ideal rings, particularly chain rings; EGPWs and extended poset profiles are defined for modules with a composition series, and Wei-type duality is established over arbitrary quasi-Frobenius rings.\"}]",1784189548,83,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"generalized-rank-weight-and-extended-generalized-poset-weight-for-codes-over-rings-a-galois-connection-approach","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/generalized-rank-weight-and-extended-generalized-poset-weight-for-codes-over-rings-a-galois-connection-approach/83655/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","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 regarding generalized weights of codes?","Question",{"text":75,"@type":76},"The paper investigates generalized rank weights and extended generalized poset weights for codes over rings, aiming to unify and reinterpret multiple coding-theoretic properties using a Galois connection approach.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the Galois connection framework relate to coding-theoretic properties?",{"text":80,"@type":76},"It reformulates diverse properties—such as Singleton/MDS-type bounds and classifications of MRD and near-MRD families, as well as evasive subspace behavior—into statements expressed through Galois connections.",{"name":82,"@type":73,"acceptedAnswer":83},"Which algebraic structures are considered when extending GRWs and EGPWs?",{"text":84,"@type":76},"GRWs and rank profiles are developed for modules over principal ideal rings, particularly chain rings; EGPWs and extended poset profiles are defined for modules with a composition series, and Wei-type duality is established over arbitrary quasi-Frobenius rings.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]