[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83051-en":3,"doc-seo-83051-105":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},83051,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Unique Insertion Error Patterns in Levenshtein’s Reconstruction Problem","Levenshtein’s sequence reconstruction model supports reliable information retrieval in advanced memory systems, including DNA-based storage. A transmitted word x∈Znq is sent through N noisy channels, where errors may include substitutions, deletions, and insertions. This study focuses on insertion errors and the channel count needed to recover x with no ambiguity. Two generalized reconstruction models are analyzed—multiset and non-multiset—allowing identical outputs from different insertion patterns. The work derives Nmq(n,1) and Nnmq(n,1), proves their equality for all n and q, characterizes optimal attaining words, and provides bounds and recursive construction results.","arXiv :2607 .06 18 1v 1 [ cs .IT] 7 Jul 2026  \nUnique Insertion Error Patterns in Levenshtein’s  \nReconstruction Problem  \nVille Junnila, Tero Laihonen, Tuomo Lehtil and Pavan Padavu Devaraj Department of Mathematics and Statistics  \nUniversity of Turku, FI-20014 Turku, Finland  \nEmail: {viljun, terolai, tualeh}@utu.fi, pavanpdevaraj@gmail.com  \nAbstract  \nLevenshtein’s sequence reconstruction model plays an essential role in information retrieval of advanced memory systems, such as the DNA-based storage systems. In the Levenshtein’s model, a word x ∈ Znq is transmitted through N noisy channels, and the goal is to recover, using the output words produced by these channels, the original word x unambiguously, or with small uncertainty L. Errors occurring in the channels usually involve substitutions, insertions and deletions. In this work, we focus on insertion errors. One of the main questions in this context is determining the minimum number of channels N required to recover the transmitted word x. The original formulation of Levenshtein’s sequence reconstruction problem requires that all the output words from the channels are distinct. However, channels may produce the same output word even if different insertion errors occur in them. In this paper, we investigate two reconstruction models where the channels are allowed to produce identical output words even though different insertion errors occur in the channels. These two models, called the multiset model and non-multiset model, generalize the Levenshtein’s model. Let us denote the minimum number of channels required to unambiguously recover the transmitted word x ∈ Znq by Nmq(n, t) + 1 in the multiset model and Nnmq(n, t) + 1 in the non-multiset model, where t denotes the exact number of insertions occurring in a channel. We determine Nmq(n,1) and Nnmq(n,1) for all n and q, and show the somewhat surprising fact that Nmq(n, 1) = Nnmq(n,1) . Moreover, we provide a full characterization of the words that attain this value. We also give a general lower bound on Nmq(n, t) for t ≥ 1 and a recursive upper bound. For t = 1, we consider a construction from codes C ⊆ Znq to codes C′ ⊆ Znq+2 such that the number of channels required to determine the transmitted word x ∈ C′ is small. This construction is shown to be optimal for certain parameters.  \nKeywords: Levenshtein’s Sequence Reconstruction, Information Retrieval, Insertion Errors, Different Error Patterns, DNA Storage.  \nI. INTRODUCTION  \nLevenshtein’s sequence reconstruction problem, introduced in [2], has gained renewed attention due to its relevance in information retrieval for advanced storage technologies, such as DNA based ones [3],[4] . In the information retrieval process of DNA data storage (see [5]–[8]), numerous copies of the stored information are obtained, each typically affected by substitution, deletion, and insertion errors. The goal is to recover the original information using these erroneous copies. For results on this problem, see, for example,[2], [3], [9]–[18] .  \nLet us first introduce some notation. We represent the set {a, a+1,..., b} by [a, b] for integers a ≤ b. Let Zq = {0, 1 , . . . , q − 1} denote the ring of q ≥ 2 elements, and Znq = Zq × · · · × Zq (n times) . For a word x = x 1 . . . xn ∈ Znq, we denote by x [a,b] the shortened subword xaxa+1 . . . xb ∈ Zbq−a+1 of x. The all-zero word 00 ... 0 ∈ Znq is denoted by 0 and the empty word by ε . The Hamming weight w (x) of x ∈ Znq is the number of non-zero coordinates of x. A non-empty subset of Znq is called a code and its elements are called codewords. For a set A, the notation |A| is the usual cardinality of the set. Given a multiset A, that is, a collection of elements in which elements can be repeated multiple times, let set(A) denote the set of distinct words in A and let m(a, A) denote the multiplicity of a in the multiset A. If A is a multiset, then by |A| we denote the total cardinality of the multiset, that is, the sum of multiplicities in A","cbCaihOakRzn7voS","https://ap.wps.com/l/cbCaihOakRzn7voS","pdf",462498,3,1,22,"English","en",105,"# Introduction\n# Notation and Traditional Model\n# Channel Model and Output Ambiguity\n# Main Contributions and Results","[{\"question\":\"What is Levenshtein’s sequence reconstruction problem in this work?\",\"answer\":\"A word x∈Znq is transmitted through N channels that introduce bounded errors, and the receiver attempts to recover x unambiguously (or with small uncertainty) from the observed outputs.\"},{\"question\":\"How do the multiset and non-multiset reconstruction models differ?\",\"answer\":\"Both generalize Levenshtein’s model by allowing different insertion error patterns to produce identical output words. The distinction lies in how identical outputs are treated under multiset versus non-multiset assumptions.\"},{\"question\":\"What key result is shown for the case of exactly one insertion per channel?\",\"answer\":\"For t=1, the minimum required channel counts satisfy Nmq(n,1)=Nnmq(n,1) for all n and q, and the paper provides a full characterization of the words achieving this value.\"}]",1784184880,55,{"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},"unique-insertion-error-patterns-in-levenshteins-reconstruction-problem","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/unique-insertion-error-patterns-in-levenshteins-reconstruction-problem/83051/",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","2026-07-23","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 is Levenshtein’s sequence reconstruction problem in this work?","Question",{"text":75,"@type":76},"A word x∈Znq is transmitted through N channels that introduce bounded errors, and the receiver attempts to recover x unambiguously (or with small uncertainty) from the observed outputs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the multiset and non-multiset reconstruction models differ?",{"text":80,"@type":76},"Both generalize Levenshtein’s model by allowing different insertion error patterns to produce identical output words. The distinction lies in how identical outputs are treated under multiset versus non-multiset assumptions.",{"name":82,"@type":73,"acceptedAnswer":83},"What key result is shown for the case of exactly one insertion per channel?",{"text":84,"@type":76},"For t=1, the minimum required channel counts satisfy Nmq(n,1)=Nnmq(n,1) for all n and q, and the paper provides a full characterization of the words achieving this value.","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 & Novel",90,"story-novel",{"id":47,"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":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]