[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84339-en":3,"doc-seo-84339-105":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},84339,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","On the Etzion-Silberstein Conjecture for Block Ferrers Diagrams","Ferrers diagram rank-metric codes with prescribed support have dimension bounded above by the Etzion–Silberstein (ES) bound. This paper studies this question for block Ferrers diagrams, where dots form fixed-size square blocks. It introduces MSRD-constructibility by replacing diagonal MDS codes with maximum sum-rank distance (MSRD) codes on block diagonals. It proves that MSRD-constructible pairs yield optimal codes over sufficiently large finite fields, relates block-constructibility to contraction via lifting criteria, derives cases for strictly block-monotone and initially block-convex diagrams, and reduces to block triangular diagrams to obtain new arbitrary-field results.","arXiv :2607 .08239v1 [math .CO] 9 Jul 2026  \nOn the Etzion-Silberstein conjecture for block Ferrers diagrams  \nMarco Calderini Marta Messia Alessandro Neri  \nAbstract  \nFerrers diagram rank-metric codes are rank-metric codes with prescribed support, and their dimension is bounded from above by the Etzion–Silberstein bound. In this paper, we study this problem for block Ferrers diagrams, namely Ferrers diagrams whose dots are grouped into square blocks of a fixed size. Motivated by the diagonal construction for MDS-constructible Ferrers diagrams, we introduce the notion of MSRD-constructibility, where MDS codes on diagonals are replaced by maximum sum-rank distance (MSRD) codes on block diagonals. We show that MSRDconstructible pairs yield optimal Ferrers diagram rank-metric codes over sufficiently large finite fields. We then relate MSRD-constructibility of a block Ferrers diagram to MDS-constructibility of its contraction, proving an equivalence when the distance is compatible with the block size and giving lifting criteria in the general case. As a consequence, we obtain MSRD-constructibility for strictly block-monotone and initially block-convex diagrams. Finally, we prove a reduction to block triangular diagrams and use it to obtain new arbitrary-field cases of the Etzion–Silberstein conjecture for MSRD-constructible block Ferrers diagrams.  \nKeywords: Rank-metric codes, block Ferrers diagrams, Etzion-Silberstein conjecture, MSRDconstructible diagram  \n1 Introduction  \nContext  \nRank-metric codes were introduced by Delsarte [5] and later studied by Gabidulin [8] . Since then, they have become a central object in coding theory, partly because of their applications to network coding [21], crisscross error correction [20], and cryptography [9] . A particularly important class of rank-metric codes is given by maximum rank distance codes, or MRD codes, namely codes attaining the Singleton-like bound for the rank metric.  \nFerrers diagram rank-metric codes arise naturally in the construction of constant-dimension subspace codes. In the multilevel construction proposed by Etzion and Silberstein [7], one first fixes a Schubert cell in a Grassmannian. The pivot positions defining the Schubert cell determine a Ferrers diagram, and the matrices parameterizing the points in that cell are precisely matrices supported on this diagram. Thus, within each Schubert cell, the problem of constructing large subspace codes leads to the problem of constructing rank-metric codes with a prescribed Ferrers-diagram support. Given a Ferrers diagram F, an [F, k, d]q Ferrers diagram rank-metric code is a rank-metric code whose codewords are supported on F. Etzion and Silberstein proved a Singleton-like upper bound for the  \nM. Calderini: Department of Mathematics, University of Trento, Trento, Italy; e-mail: marco.calderini@unitn.it  \nM. Messia: Department of Mathematics and Data Science, Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussel, Belgium; [e-mail](e-mail: marta.messia@vub.be)[: marta.messia@vub.be](e-mail: marta.messia@vub.be)  \nA. Neri: Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, Naples, Italy; e-mail: [alessandro.neri@unina.it](alessandro.neri@unina.it)  \nMathematics Subject Classification (2020): Primary 11T71; Secondary 94B05  \ndimension of such codes [7] . More precisely, if the minimum rank distance is d, then the dimension of any [F, k, d]q code is at most νmin (F, d), a combinatorial value obtained from the diagram F by erasing d − 1 between rows and columns. Thus, throughout the paper, νmin (F, d) denotes precisely the Etzion–Silberstein upper bound. Etzion and Silberstein conjectured that this bound is always attainable: for every Ferrers diagram F of order n, every 1 ≤ d ≤ n, and every finite field F q , there exists an [F, νmin (F, d), d]q maximum Ferrers diagram rank-metric code.  \nDespite substantial progress, the Etzion–Silberstein conjecture remains open in full generality","cbCaiu8yLcO6wpab","https://ap.wps.com/l/cbCaiu8yLcO6wpab","pdf",571680,2,1,33,"English","en",105,"# Introduction\n## Rank-metric codes and Ferrers-diagram rank-metric codes\n## Etzion–Silberstein conjecture and known approaches\n## Diagonal constructions and new MSRD-constructibility concept","[{\"question\":\"What problem does the Etzion–Silberstein conjecture address for Ferrers diagram rank-metric codes?\",\"answer\":\"It conjectures that the Etzion–Silberstein upper bound on the dimension of an [F,k,d]q Ferrers-diagram rank-metric code is always attainable for any Ferrers diagram F, distance d, and finite field size q.\"},{\"question\":\"How does the paper extend diagonal MDS constructions to block Ferrers diagrams?\",\"answer\":\"It replaces placing MDS codes on the Hamming diagonals by placing MSRD (maximum sum-rank distance) codes on block diagonals, leading to the notion of MSRD-constructibility.\"},{\"question\":\"What is the relationship between MSRD-constructibility of a block diagram and MDS-constructibility of its contraction?\",\"answer\":\"The paper proves an equivalence when the distance is compatible with the block size, and provides lifting criteria in the general case to transfer constructibility between the block diagram and its contraction.\"}]",1784194921,83,{"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},"on-the-etzion-silberstein-conjecture-for-block-ferrers-diagrams","",{"@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/on-the-etzion-silberstein-conjecture-for-block-ferrers-diagrams/84339/",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","2026-07-24","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 Etzion–Silberstein conjecture address for Ferrers diagram rank-metric codes?","Question",{"text":75,"@type":76},"It conjectures that the Etzion–Silberstein upper bound on the dimension of an [F,k,d]q Ferrers-diagram rank-metric code is always attainable for any Ferrers diagram F, distance d, and finite field size q.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper extend diagonal MDS constructions to block Ferrers diagrams?",{"text":80,"@type":76},"It replaces placing MDS codes on the Hamming diagonals by placing MSRD (maximum sum-rank distance) codes on block diagonals, leading to the notion of MSRD-constructibility.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the relationship between MSRD-constructibility of a block diagram and MDS-constructibility of its contraction?",{"text":84,"@type":76},"The paper proves an equivalence when the distance is compatible with the block size, and provides lifting criteria in the general case to transfer constructibility between the block diagram and its contraction.","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},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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