[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86002-en":3,"doc-seo-86002-105":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},86002,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Wei-type Duality and Asymptotics of the Footprint Bound","This work establishes a Wei-type duality linking the footprint bound and the dual footprint bound for generalized Hamming weights of evaluation codes. The duality also connects the Andersen-Geil bound with the Feng-Rao bound. A further result shows that neither the footprint bound nor its dual version can certify asymptotic goodness for families of evaluation codes, as the corresponding normalized footprint bounds vanish in the asymptotic regime.","arXiv :2607 . 10680v 1 [ cs .IT] 12 Jul 2026  \nWEI-TYPE DUALITY AND ASYMPTOTICS OF THE FOOTPRINT BOUND  \nRODRIGO SAN-JOS´E  \nAbstract. We obtain a Wei-type duality between the footprint bound and the dual footprint bound for the generalized Hamming weights of an evaluation code.  \nThis duality applies between the Andersen-Geil and Feng-Rao bounds as well.  \nWe also prove that the footprint and dual footprint bounds cannot be used to guarantee the asymptotic goodness of a family of evaluation codes.  \n1. Introduction  \nThe footprint bound is a classical result in algebraic geometry, bounding the number of rational points defined by an ideal in terms of its footprint [10, Prop. 7, Chapter 5 §3] . This idea has proven fruitful for bounding the minimum distance of linear codes [19, 41, 43], since it is closely related to the number of zeroes of polynomials over a finite field. The generalized Hamming weights (GHWs) of alinear code, introduced in [48], provide an extension of the minimum distance, which has found several applications [28, 29] . Moreover, it is related to the number of common zeroes of sets of polynomials and can therefore also be studied using the footprint bound. Among other properties, the GHWs of a linear code satisfy the so-called Wei duality, which implies that the GHWs of a linear code are determined by those of its dual, and vice versa. A further extension of this concept is given by the relative generalized Hamming weights (RGHWs) of a pair of linear codes [39], which have applications in secret sharing [35] and quantum error-correction [30,33] . There are two related bounds, the Feng-Rao bound [14], and the AndersenGeil bound (sometimes called the Feng-Rao bound for primary codes) [1], which generalize the footprint bound in the affine setting and can also be used to bound the RGHWs of linear codes [23] . The computation of the GHWs and RGHWs of linear codes is, in general, NP-hard [47], and the footprint and Feng-Rao-type bounds have been used to obtain them for many of the most well-known families of codes [2,3,6,7,9,11,21,23,31] .  \nFinding constructions for asymptotically good families of codes has proven a challenging problem. In fact, many of the most well-known algebraic families of codes are known to be asymptotically bad, e.g. , Reed-Muller codes, Cartesian codes, hyper  \nbolic codes  [16], binary primitive narrow-sense BCH codes [36], and several classes 2020 Mathematics Subject Classification. Primary: 94B05 . Secondary: 11T71, 14G50 .  \nKey words and phrases. Footprint bound, evaluation codes, generalized Hamming weights.  \nThe author was partially supported by the NSF grant DMS-2401558, the Commonwealth Cyber Initiative, an AMS-Simons Travel Grant, and by Grant PID2022-138906NB-C21 funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU.  \n2 RODRIGO SAN-JOS´E  \nof cyclic codes [4,42] . One of the most important examples of asymptotically good families of codes is given by AG codes [15, 46] . The authors consider the Goppa bound to prove that the corresponding codes are asymptotically good. Since the Feng-Rao and Andersen-Geil bounds are stronger, these bounds can be used to certify asymptotic goodness, in the sense that one can prove that certain families of codes are asymptotically good using them instead of the actual minimum distance of the code.  \nIn this paper, we study the footprint bound and the dual footprint bound from a combinatorial perspective. In particular, in Section 3 we prove that there is a Wei-type duality between the values of the footprint bound and the dual footprint bound, and, more generally, between the values of the Andersen-Geil bound and the Feng-Rao bound. This implies that the footprint bound is sharp if and only if the dual footprint bound is sharp. Moreover, we show that the dual footprint bound coincides with the footprint bound of the dual code whenever the dual code is also obtained as a monomial code via an order-reversing bijection. This covers several wel","cbCaimMtkshc7hSk","https://ap.wps.com/l/cbCaimMtkshc7hSk","pdf",352344,3,1,16,"English","en",105,"# Introduction\n## Footprint bound and generalized Hamming weights\n## Wei duality and related bounds\n# Preliminaries\n## Finite fields and linear codes\n## Generalized Hamming weights","[{\"question\":\"What duality does the paper prove between bounds?\",\"answer\":\"It proves a Wei-type duality between the footprint bound and the dual footprint bound for generalized Hamming weights of an evaluation code, and also between the Andersen-Geil bound and the Feng-Rao bound.\"},{\"question\":\"How does the paper relate these bounds to asymptotic goodness?\",\"answer\":\"It proves that the footprint bound and the dual footprint bound cannot be used to guarantee asymptotic goodness of a family of evaluation codes; the normalized footprint bound quotient vanishes asymptotically.\"},{\"question\":\"What are the generalized Hamming weights (GHWs) studied here?\",\"answer\":\"For an [n,k] linear code, the r-th GHW is defined as the minimum size of the support among r-dimensional subcodes, extending the minimum distance concept.\"}]",1784207691,40,{"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},"wei-type-duality-and-asymptotics-of-the-footprint-bound","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/wei-type-duality-and-asymptotics-of-the-footprint-bound/86002/",4,{"url":51,"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 duality does the paper prove between bounds?","Question",{"text":75,"@type":76},"It proves a Wei-type duality between the footprint bound and the dual footprint bound for generalized Hamming weights of an evaluation code, and also between the Andersen-Geil bound and the Feng-Rao bound.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper relate these bounds to asymptotic goodness?",{"text":80,"@type":76},"It proves that the footprint bound and the dual footprint bound cannot be used to guarantee asymptotic goodness of a family of evaluation codes; the normalized footprint bound quotient vanishes asymptotically.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the generalized Hamming weights (GHWs) studied here?",{"text":84,"@type":76},"For an [n,k] linear code, the r-th GHW is defined as the minimum size of the support among r-dimensional subcodes, extending the minimum distance concept.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"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":52,"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":29,"slug":118},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]