[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86026-en":3,"doc-seo-86026-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86026,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Minimum Distance and Decoding of Coxeter Codes","A binary Coxeter code CW(r) associated with a finite Coxeter system (W,S) is defined as the F2-linear span of indicator vectors of standard cosets of a fixed rank. Coxeter codes generalize Reed–Muller codes, and prior work conjectured an exact minimum distance formula. This paper proves the conjecture by showing the distance equals the order of the smallest parabolic subgroup of the corresponding rank. It further extends Reed’s majority-logic decoding via a Coxeter-theoretic shadow-code approach.","arXiv :2607 . 10774v 1 [ cs .IT] 12 Jul 2026  \nMINIMUM DISTANCE AND DECODING OF COXETER CODES  \nALEXANDER BARG, Q¨ENDRIM R. GASHI, AND TIANYUAN XU  \nAbstract. A binary Coxeter code associated with a finite Coxeter system (W, S) is an F2-linear span of indicators of standard cosets of a fixed rank. Coxeter codes, introduced in a recent paper by N. Coble and A. Barg, are a generalization of Reed–Muller codes which arise when W = Zm2 is the Coxeter group of type mA 1 . In that paper, the authors proposed a conjectural value for the minimum distance of a general Coxeter code. This conjecture is proved in the present work. As a consequence, we obtain a Coxeter-theoretic generalization of Reed’s majority-logic decoding algorithm for Reed–Muller codes.  \n1. Introduction  \nLet (W, S) be a finite Coxeter system, where W is a finite group with a generating set S = {s1 ,..., sm} satisfying the defining relations (sisj)M (i,j) = 1 with M(i, i) = 1 and M(i, j) = M (j, i) ≥ 2 for i  j. A standard (left) coset of rank r in W is a coset of the form wWI, where w ∈ W, I ⊆ S, and WI is a standard parabolic subgroup of rank |I| = r. We refer to [4] for an introduction to the combinatorics of Coxeter groups.  \nRecently, Coble and Barg [6] introduced a class of binary linear codes spanned by indicators of standard cosets of W of a fixed rank. Specifically, they defined a Coxeter code CW(r) of order r ∈ {−1, 0 ,..., m} to be the F2-linear span of indicator vectors of the standard cosets of rankm − r:  \n(1) CW(r) := Span{1wWI | w ∈ W, I ⊆ S,|I| = m − r} .  \nThe value −1 is included for convenience since it enables one to properly formulate the duality results for Coxeter codes [6, Sec.3] . At the same time, we have CW(−1) = {0} by definition. The codes in (1) form a direct generalization of a classic family of binary codes known as Reed–Muller codes RM(r, m), which correspond to Coxeter type mA 1 , where S = {e1 ,..., em} and W = Zm2 is a direct product of m permutation groups on 2 elements, with generators corresponding to the vectors of the standard basis. Reed–Muller codes have been extensively studied in classical coding theory [10, Ch.13–15], [2], [1], and they also give rise to a family of quantum CSS codes with a range of well-understood properties [3] .  \nThe Coxeter code CW(r) has length |W| by construction, and its dimension is  \n(2)  \nrdim CW(r) =X  \ni=0  \n􀀜 Wi􀀝 ,  \nwhere the W-Eulerian number 􀀊 Wi􀀋, i ∈ {0,..., m} is the count of elements in W with descent number equal to i, [4, Sec.7.2] . For the classical Reed–Muller code RM (r, m), this dimension reduces to the familiar expression Pri=0 􀀀mi􀀁; see [6] for a proof for general W.  \nIn addition to length and dimension, the third important parameter of a code C is its distance dist(C), which equals the minimum Hamming distance between two distinct codewords. Fora linear code, one has dist(C) = minx∈C\\{0}wt(x), where wt denotes the Hamming weight. The classical Reed–Muller codes are well known to have distance dist(RM(r, m)) = 2m−r [10, Thm.13.3] . For a general Coxeter code CW(r), it is clear from (1) that dist(CW(r)) is at most  \n2 A. BARG, Q. R. GASHI, AND T. XU  \nthe size of the smallest standard parabolic subgroup of rank m − r. Coble and Barg conjectured that this upper bound is in fact the exact distance for general Coxeter codes. In this paper, we prove this conjecture:  \nTheorem 1.1 . Let (W, S) be a finite Coxeter system of rank m and let dr := minI⊆S,|I| =m−r|WI| be the order of the smallest parabolic subgroup of rank m − r. Then  \ndist(CW(r)) = dr .  \nOur proof of Theorem 1.1 proceeds by reduction to certain projections of the code CW(r) which we call shadow codes, and which we explore on their own merit, finding their bases and parameters. The core part of the proof is a lower bound on the distance of a shadow code, which is then taken back to the main code by peeling off fixed-rank layers starting from the top level. While our arguments are phrased in combinatorial terms, the k","cbCaitau6IF0Zwcm","https://ap.wps.com/l/cbCaitau6IF0Zwcm","pdf",378659,5,1,16,"English","en",105,"# Introduction\n# Preliminaries","[{\"question\":\"How are Coxeter codes CW(r) defined in this work?\",\"answer\":\"CW(r) is the F2-linear span of indicator vectors of standard cosets of a fixed rank in a finite Coxeter system (W,S). The cosets are determined by standard parabolic subgroups associated with subsets of S.\"},{\"question\":\"What is the main result about the minimum distance of CW(r)?\",\"answer\":\"For a finite Coxeter system of rank m and dr defined using the smallest parabolic subgroup of rank m−r, the paper proves dist(CW(r)) = dr.\"},{\"question\":\"How does the paper relate the decoding problem to Reed–Muller codes?\",\"answer\":\"The proof uses shadow codes to obtain a distance lower bound, and the resulting ideas are adapted to produce a decoding algorithm generalizing Reed’s majority-logic decoding for Reed–Muller codes.\"}]",1784207896,40,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"minimum-distance-and-decoding-of-coxeter-codes","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/minimum-distance-and-decoding-of-coxeter-codes/86026/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"How are Coxeter codes CW(r) defined in this work?","Question",{"text":76,"@type":77},"CW(r) is the F2-linear span of indicator vectors of standard cosets of a fixed rank in a finite Coxeter system (W,S). The cosets are determined by standard parabolic subgroups associated with subsets of S.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the main result about the minimum distance of CW(r)?",{"text":81,"@type":77},"For a finite Coxeter system of rank m and dr defined using the smallest parabolic subgroup of rank m−r, the paper proves dist(CW(r)) = dr.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the paper relate the decoding problem to Reed–Muller codes?",{"text":85,"@type":77},"The proof uses shadow codes to obtain a distance lower bound, and the resulting ideas are adapted to produce a decoding algorithm generalizing Reed’s majority-logic decoding for Reed–Muller codes.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"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":20,"slug":137},19,"General","general"]