[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83247-en":3,"doc-seo-83247-105":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},83247,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Combinatorial Constructions of Schubert Subspace Codes","Schubert subspace codes are constant-dimension subspace codes with prescribed intersection behavior relative to a fixed subspace. The work aims to build codes of maximum size in extremal distance regimes where a natural counting upper bound holds. Two construction families are developed: one via direct-sum decomposition, partial spreads, and colorings of q-Johnson graphs, with necessary chromatic and clique obstruction criteria; the other via field reduction from evasive and scattered subspaces, yielding exact sizes in the scattered case and recovering a prior construction.","arXiv :2607 .07479v2 [math .CO] 9 Jul 2026  \nCOMBINATORIAL CONSTRUCTIONS OF SCHUBERT SUBSPACE CODES  \nGIANIRA N. ALFARANO 1 , ALESSANDRO NERI2 , AND BEATRICE TOESCA3  \nAbstract. We study Schubert subspace codes, which are constant-dimension subspace codes with prescribed intersection conditions with a fixed subspace. Our goal is to construct codes of maximum possible size in the extremal distance cases where a natural counting upper bound applies. We give two families of constructions. The first one uses a direct-sum decomposition of the ambient space, together with partial spreads and colorings of powers of q-Johnson graphs.  \nFor this construction, we also prove necessary conditions, which show how chromatic and clique obstructions arise. The second family is obtained by field reduction from evasive and scattered subspaces over extension fields. This gives codes whose size can be computed exactly in the scattered case and recovers the only previously known construction as a special case.  \nKeywords: Schubert subspace codes; constant-dimension codes; partial spreads; q-Johnson graphs; scattered subspaces; field reduction.  \nMSC (2020): 94B05, 94B27, 05C15, 51E20  \n1. Introduction  \nSubspace codes were introduced in the context of random network coding as a natural errorcorrecting framework for noncoherent communication over networks. In this model, intermediate nodes are allowed to perform linear combinations of incoming messages before forwarding them. It is natural in the area of network coding to consider the elements of the Grassmannian Gr q(k, n) as codewords of a so-called subspace code. The foundational work of K¨otter and Kschischang initiated the systematic study of such codes and introduced several basic bounds and constructions [18] . Since then, subspace codes, and in particular constant-dimension codes, have been studied extensively; see, for instance, [17, 19] and references therein. Constant-dimension codes are subsets of the Grassmannian Grq(k, n), and the subspace distance between two codewords U, V ∈ Grq(k, n) is given by dS (U, V ) = 2(k − dim(U ∩ V )) . Thus, constructing large constantdimension codes with a prescribed minimum distance is equivalent to constructing large families of k-dimensional subspaces with controlled pairwise intersections.  \nSeveral important constructions of constant-dimension codes are based on rank-metric codes. The lifting of rank-metric codes, and in particular of maximum rank-distance codes, produces large families of subspaces with good distance properties [18] . This point of view also connects the theory of subspace codes with the geometry of Grassmannians, where subspaces are represented by row spaces of matrices in reduced row echelon form. In this direction, Ferrers diagram rankmetric codes and multilevel constructions provide flexible methods for building large codes; see [10] . These constructions show that imposing geometric or combinatorial restrictions on the support of codewords can still lead to large and structured families of subspaces.  \nIn this paper, we study Schubert subspace codes, which were recently introduced in [1] . These are subspace codes with an additional geometric constraint: the codewords are required to lie ina fixed Schubert variety of the Grassmannian. Schubert varieties are among the most classical subvarieties of Grassmannians and they are defined by imposing lower bounds on the dimensions of the intersections with the subspaces in a fixed flag. Classical references for Schubert varieties over arbitrary fields include [11, 16] . In the coding-theoretic setting, Schubert varieties over finite  \n1 Universit´e de Rennes, IRMAR, Campus de Beaulieu, F-35042 Rennes Cedex, France.  \n2 Department of Mathematics and Applications “R. Caccioppoli”, University of Naples Federico II, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy.  \n3 Institute of Mathematics, University of Zurich, Switzerland.  \nE-mail addresses: [gianira-nicoletta.alfarano@univ-rennes","cbCaiuGXaqk0pN8c","https://ap.wps.com/l/cbCaiuGXaqk0pN8c","pdf",357619,2,1,20,"English","en",105,"# Introduction\n## Subspace codes and constant-dimension framework\n## Schubert subspace codes and incidence constraints\n## Prior work and the (ℓ,t)-intersecting viewpoint","[{\"question\":\"What are Schubert subspace codes, and how do they differ from general subspace codes?\",\"answer\":\"They are constant-dimension subspace codes whose codewords are required to lie in a fixed Schubert variety, imposing incidence/intersection constraints with a predetermined subspace. This converts the coding construction into an incidence-geometry problem.\"},{\"question\":\"What optimization goal does the paper pursue for Schubert subspace codes?\",\"answer\":\"To construct codes of maximum possible size in extremal distance cases where a natural counting upper bound applies, and to determine when that bound is attained.\"},{\"question\":\"What are the two main construction approaches presented?\",\"answer\":\"One family uses a direct-sum decomposition together with partial spreads and colorings of q-Johnson graphs, including proofs of necessary conditions from chromatic/clique obstructions. The second family uses field reduction from evasive and scattered subspaces over extension fields, enabling exact size computation in the scattered case and matching a previously known construction as a special case.\"}]",1784186231,50,{"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},"combinatorial-constructions-of-schubert-subspace-codes","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/combinatorial-constructions-of-schubert-subspace-codes/83247/",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","2026-07-22","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 are Schubert subspace codes, and how do they differ from general subspace codes?","Question",{"text":75,"@type":76},"They are constant-dimension subspace codes whose codewords are required to lie in a fixed Schubert variety, imposing incidence/intersection constraints with a predetermined subspace. This converts the coding construction into an incidence-geometry problem.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What optimization goal does the paper pursue for Schubert subspace codes?",{"text":80,"@type":76},"To construct codes of maximum possible size in extremal distance cases where a natural counting upper bound applies, and to determine when that bound is attained.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the two main construction approaches presented?",{"text":84,"@type":76},"One family uses a direct-sum decomposition together with partial spreads and colorings of q-Johnson graphs, including proofs of necessary conditions from chromatic/clique obstructions. The second family uses field reduction from evasive and scattered subspaces over extension fields, enabling exact size computation in the scattered case and matching a previously known construction as a special case.","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},[93,97,101,105,110,114,119,122,126,129,133],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"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":29,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"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":22,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":127,"show_sort_weight":22,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":106,"slug":136},19,"General","general"]