[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83660-en":3,"doc-seo-83660-105":30,"detail-sidebar-cat-0-en-105":96},{"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},83660,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","The Weight Distribution of the Third-Order Reed-Muller Code of Length 2048","Computation and analysis of the weight distribution of the third-order Reed–Muller code RM(3,11) of length 2048. The weight enumerator is assembled from coset weight enumerators of f+RM(2,10), evaluated using representatives of 3,691,560 nonzero GL(10,2)-orbits of Boolean cubic forms. A structural theorem enables nondegenerate hyperplane restrictions, accelerating enumeration and yielding second-order nonlinearity values. Results improve the covering-radius bound for RM(2,10) from 400 to 408 and refine related bounds for RM(6,10) vs. RM(7,10).","The Weight Distribution of the Third-Order Reed–Muller Code of Length 2048  \nKirill Khoruzhii1 ,∗ , Patrick Gelß1 , Sebastian Pokutta1 ,2  \n1 Zuse Institute Berlin, Berlin, Germany  \n2 Technische Universität Berlin, Germany  \nWe compute the weight distribution of the third-order Reed–Muller code RM(3 , 11) of length  \n2048. The weight enumerator is assembled from the coset weight enumerators of f + RM(2 , 10) , evaluated for representatives of all 3691560 nonzero GL(10 , 2)-orbits of Boolean cubic forms in ten variables. The computation rests on a structural theorem: a nondegenerate Boolean cubic form admits a nondegenerate hyperplane restriction, except for a single orbit in each odd dimension. The same pass determines the second-order nonlinearity of every cubic form: the relative covering radius of RM(2 , 10) in RM(3 , 10) is 408, attained on 179 orbits. This raises the best known lower bound on the covering radius of RM(2, 10) from 400 to 408. A complementary heuristic search shows that the relative covering radius of RM(6, 10) in RM(7 , 10) is at most 32, improving the previous bound of 50.  \narXiv :2607 .02365v 1 [ cs .IT] 2 Jul 2026  \n1. Introduction  \nThe weight distributions of the Reed–Muller codes RM(r, m) [16, 17] are a classical subject of coding theory, yet they are known in closed form only for orders r ⩽ 2 [20] and, through the MacWilliams identities, for the dual orders r ⩾ m − 3 (see [1] for a survey) . Beyond these layers, general results cover only the lowweight range [11], and complete distributions are known only from individual computations. For the third-order codes the known cases form a chain of length-doubling steps: RM(3 , 7) [21], RM(3, 8) [18, 23], RM(3, 9) [22], and RM(3 , 10) [3]; recently the fourth-order code RM(4, 9) of length 512 was enumerated [15] . This paper computes the next step of the chain, the weight distribution of the third-order Reed–Muller code RM(3 , 11) of length 2048 (Tab. 1) .  \nThe route is the Sarwate-type recursion [15, 18]: the weight enumerator of RM(3, m+1) is the sum of squared coset weight enumerators of f + RM(2, m) over all cubic forms f, and since a coset enumerator depends only on the GL(m,2)-orbit of f, the sum collapses to orbit representatives weighted by orbit sizes (Sec. 2) . Every computation in the chain therefore rests on a classification one dimension lower: Hou classified the cubic forms for m ⩽ 8 [10], the case m = 9 was classified in [3, 8], and the fourth-order computation [15] uses the classifications of quartic forms in eight variables and of the cosets of RM(2 , 7) in RM(4 , 7) [7] . For m = 10 the required input became available only recently: the classification of Boolean cubic forms in ten variables [13], which provides representatives and stabilizer orders for all 3691560 nonzero GL(10 , 2)-orbits.  \nThe classification alone does not make the computation feasible; the concluding remarks of [15] name the percoset cost as the principal obstacle to any progress beyond length 512. The split formula of Brier and Langevin [3] evaluates one coset enumerator by an outer enumeration over the homogeneous quadratic forms one dimension lower, 236 forms for m = 10, which is out of reach at the scale of the catalog. The second ingredient of this paper  \n∗ [khoruzhii@zib.de](khoruzhii@zib.de)  \nis a structural theorem: every nondegenerate Boolean cubic form admits a nondegenerate hyperplane restriction, except for one orbit in each odd dimension (Thm. 1) . Splitting along such a restriction saves a factor of 2m in the enumeration, 226 instead of 236 cosets per orbit atm = 10 . With this speedup the full catalog pass takes about 65 CPU-years and yields the coset weight enumerator of every orbit (Sec. 3) .  \nThe lowest nonzero coefficient of the coset enumerator of f is its second-order nonlinearity d2 (f), a longstudied quantity in the covering-radius literature [9] and in cryptographic Boolean function analysis, and more recently a measure of Clifford approx","cbCaihNixrd9da2a","https://ap.wps.com/l/cbCaihNixrd9da2a","pdf",620556,5,1,9,"English","en",105,"# Introduction\n## Problem statement and motivation\n## Sarwate-type recursion and orbit classification\n## Structural theorem and computational speedup\n## Covering radius and accompanying results\n# Reed–Muller Codes\n## Definitions and key quantities","[{\"question\":\"What code and length does the document focus on?\",\"answer\":\"The document computes the weight distribution of the third-order Reed–Muller code RM(3,11) with length 2048.\"},{\"question\":\"How is the weight enumerator of RM(3,11) obtained?\",\"answer\":\"It is assembled from coset weight enumerators of f+RM(2,10), using representatives of GL(10,2)-orbits of Boolean cubic forms and collapsing the sum by orbit sizes.\"},{\"question\":\"What structural theorem makes the computation feasible?\",\"answer\":\"A structural theorem states that a nondegenerate Boolean cubic form admits a nondegenerate hyperplane restriction, except for a single orbit in each odd dimension, enabling a large enumeration speedup.\"},{\"question\":\"Which bounds are improved by the results?\",\"answer\":\"The paper raises the best known lower bound on the covering radius of RM(2,10) from 400 to 408 and, via a complementary local-search heuristic, shows an upper bound improvement for the relative covering radius of RM(6,10) in RM(7,10) up to at most 32 (improving the previous bound of 50).\"}]",1784189581,23,{"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":91,"head_meta":93,"extra_data":95,"updated_unix":28},"the-weight-distribution-of-the-third-order-reed-muller-code-of-length-2048","",{"@graph":36,"@context":90},[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/the-weight-distribution-of-the-third-order-reed-muller-code-of-length-2048/83660/",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","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82,86],{"name":73,"@type":74,"acceptedAnswer":75},"What code and length does the document focus on?","Question",{"text":76,"@type":77},"The document computes the weight distribution of the third-order Reed–Muller code RM(3,11) with length 2048.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is the weight enumerator of RM(3,11) obtained?",{"text":81,"@type":77},"It is assembled from coset weight enumerators of f+RM(2,10), using representatives of GL(10,2)-orbits of Boolean cubic forms and collapsing the sum by orbit sizes.",{"name":83,"@type":74,"acceptedAnswer":84},"What structural theorem makes the computation feasible?",{"text":85,"@type":77},"A structural theorem states that a nondegenerate Boolean cubic form admits a nondegenerate hyperplane restriction, except for a single orbit in each odd dimension, enabling a large enumeration speedup.",{"name":87,"@type":74,"acceptedAnswer":88},"Which bounds are improved by the results?",{"text":89,"@type":77},"The paper raises the best known lower bound on the covering radius of RM(2,10) from 400 to 408 and, via a complementary local-search heuristic, shows an upper bound improvement for the relative covering radius of RM(6,10) in RM(7,10) up to at most 32 (improving the previous bound of 50).","https://schema.org",{"og:url":52,"og:type":92,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":94,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":97},[98,102,106,110,114,119,124,127,131,134,138],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Story & 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