[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85484-en":3,"doc-seo-85484-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85484,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Multiset Deletion Codes: Cyclic Constructions, Bounds, and Exact Results","Study deletion-correcting codes over length-n multisets drawn from a q-ary alphabet. Provide an explicit cyclic Sidon-type construction for any alphabet size q and deletion radius t, defined via a single modular congruence modulo t(t+1)q−2+1. The scheme achieves redundancy bounded by logq(t(t+1)q−2+1) and supports linear-time online decoding for fixed q and t after finite preprocessing. Prove asymptotic balance of syndrome classes and derive exact results for q=3 and q=4, with uniqueness claims and conjectures for prime alphabets.","arXiv :2601 .05636v2 [ cs .IT] 13 Jul 2026  \nMultiset Deletion Codes: Cyclic Constructions, Bounds, and Exact Results  \nAvraham Kreindel 1 , Isaac Barouch Essayag2 , Aryeh Lev Zabokritskiy (Yohananov)3  \n1 Department of Computer Science, Reichman University, Herzliya, Israel .  \n2 Research Assistant, MIGAL – Galilee Research Institute/Tel-Hai University of Kiryat Shmona and the Galilee, Kiryat Shmona, Israel .  \n3 Department of Computer Science, MIGAL – Galilee Research Institute/Tel-Hai University of Kiryat Shmona and the Galilee, Kiryat  \nShmona, Israel .  \nContributing authors: [avrahamkreindel@gmail.com](avrahamkreindel@gmail.com) ; [isaac.es@migal.org.il](isaac.es@migal.org.il) ; [yuhanalev@telhai.ac.il](yuhanalev@telhai.ac.il) ;  \nAbstract  \nWe study deletion-correcting codes in the space of length-n multisets over a qary alphabet. We present an explicit cyclic Sidon-type construction for arbitrary alphabet size q and deletion radius t, defined by a single congruence modulo t (t+1)q−2+1 . The construction has redundancy at most logq (t(t+1)q−2+1) and admits linear-time online decoding for fixed q and t after finite preprocessing. We prove that its syndrome classes are asymptotically balanced and compare several general upper bounds. For a single deletion, we show that the natural summodulo construction is asymptotically optimal for every fixed q. We also obtain exact results for q = 3 and q = 4, including uniqueness results for optimal codes in the relevant parameter ranges, and formulate conjectures for prime alphabets.  \nKeywords: Deletion channels, multiset codes, Sidon sets, Bose–Chowla construction  \nMSC Classification: 94B25 , 94B05 , 94B60  \n1  \n1 Introduction  \nCommunication models in which the order of transmitted symbols is unreliable or irrelevant arise naturally in a variety of modern systems. Unlike classical sequence-based channels, these models preserve only the multiset of symbols, discarding positional information entirely. This abstraction was introduced and systematically developed ina series of works by Kovaˇcevi´c and collaborators [7–9], who showed that many impairments of permutation channels may be expressed as operations on symbol multiplicities inside a discrete simplex. In such settings, errors do not alter symbol positions but instead modify their counts, leading to a coding-theoretic framework fundamentally different from classical Hamming or insertion–deletion models.  \nA key motivation emphasized in [8] is the connection to permutation channels, where the transmitted sequence may undergo arbitrary reordering before reception. In several physical and biochemical systems, such as molecular communication, chemical reaction networks, and DNA-based storage architectures, the receiver observes an unordered multiset of tokens rather than a structured sequence. Deletions, duplications, and molecular losses then manifest as perturbations of multiplicities, making the multiset model a natural abstraction. Similar effects appear in ℓ ∞-limited permutation channels [12] and in coding over nonlinear combinatorial structures such as trees [22], where positional indexing is degraded or absent.  \nMultiset representations also arise naturally in practical data-management tasks, most notably in large-scale inventory auditing. Supermarkets and warehouses maintain large numbers of identical items sharing the same barcode. A physical audit observes only the multiset of remaining items, not an ordered list. Any mismatch between the expected and observed inventories, arising from unrecorded removals, operational noise, or scanning errors, appears as an unknown deletion from the true multiplicity vector. Since ordering is irrelevant, the reconstruction problem becomes that of recovering the correct multiset.  \nClassical deletion-correcting codes, beginning with Levenshtein’s seminal work [13, 14] and the Varshamov–Tenengolts codes [20], rely crucially on positional information. The decoder must identify where a de","cbCaigYH45GODMuZ","https://ap.wps.com/l/cbCaigYH45GODMuZ","pdf",605521,1,47,"English","en",105,"# Abstract\n# Introduction\n## Communication and multiset models\n## Motivations from physical and biochemical systems\n## Practical inventory auditing motivation\n## From classical sequence deletion codes to multiset deletion codes\n## Problem setup: multiset space and deletion distance\n## Redundancy and linear multiset code framework\n## Connection to Sidon-type sets and syndrome constraints","[{\"question\":\"What problem do these multiset deletion codes address?\",\"answer\":\"They address deletion correction when transmitted data is represented only as a multiset of symbols, so positional information is lost and the decoder must recover the correct multiplicities.\"},{\"question\":\"What is the main construction provided in the paper?\",\"answer\":\"The paper presents an explicit cyclic Sidon-type construction for length-n multiset codes over a q-ary alphabet with deletion radius t, defined by a single congruence modulo a quantity involving t and q.\"},{\"question\":\"What results are proven for specific alphabet sizes and for general bounds?\",\"answer\":\"It proves asymptotic properties of syndrome classes and compares general upper bounds. For a single deletion it shows asymptotic optimality of a natural summo construction for fixed q, and it gives exact results for q=3 and q=4, including uniqueness results and conjectures for prime alphabets.\"}]",1784203949,118,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"multiset-deletion-codes-cyclic-constructions-bounds-and-exact-results","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/multiset-deletion-codes-cyclic-constructions-bounds-and-exact-results/85484/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 do these multiset deletion codes address?","Question",{"text":75,"@type":76},"They address deletion correction when transmitted data is represented only as a multiset of symbols, so positional information is lost and the decoder must recover the correct multiplicities.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main construction provided in the paper?",{"text":80,"@type":76},"The paper presents an explicit cyclic Sidon-type construction for length-n multiset codes over a q-ary alphabet with deletion radius t, defined by a single congruence modulo a quantity involving t and q.",{"name":82,"@type":73,"acceptedAnswer":83},"What results are proven for specific alphabet sizes and for general bounds?",{"text":84,"@type":76},"It proves asymptotic properties of syndrome classes and compares general upper bounds. For a single deletion it shows asymptotic optimality of a natural summo construction for fixed q, and it gives exact results for q=3 and q=4, including uniqueness results and conjectures for prime alphabets.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]