[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84782-en":3,"doc-seo-84782-105":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},84782,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","New Results on Limited Magnitude Error Correcting Codes","Investigates the existence, construction, and classification of limited magnitude error-correcting codes for flash memory–type errors, using splitter sets and their links to group splittings. New nonexistence theorems are proved for quasi-perfect splitter sets, alongside a complete classification of quasi-perfect B[0, 3](n) splitter sets in both singular and nonsingular regimes. Improved lower bounds for maximal B[0, 3](q) sets are derived via Cayley-graph analysis (q prime), with existence criteria for perfect B[0, 6](q) and quasi-perfect B[−4, 4](2p) cases. A general construction framework yields infinite families of (k2, k1)-limited-magnitude cyclic b-burst-correcting codes for k1+k2≤4 and all burst lengths b, plus sufficient conditions for broader parameters. Results integrate algebraic, combinatorial, and number-theoretic techniques to advance storage-code design.","1  \narXiv :2607 .05026v 1 [ cs .IT] 6 Jul 2026  \nNew Results on Limited Magnitude Error  \nCorrecting Codes  \nZhiyu Yuan∗ , Tingting Chen†, Rongquan Feng‡ and Gennian Ge§  \n∗ School of Mathematical Sciences, Peking University, Beijing 100871, China. Email: [yzhiyu_pku@pku.edu.cn](yzhiyu_pku@pku.edu.cn)  \n†Institute of Mathematics and Interdisciplinary Sciences, Xidian University, Xi’an, 710071, China. Email:  \n[ttchenxu@mail.ustc.edu.cn](ttchenxu@mail.ustc.edu.cn)  \n‡School of Mathematical Sciences, Peking University, Beijing 100871, China. Email: [fengrq@math.pku.edu.cn](fengrq@math.pku.edu.cn)  \n§ School of Mathematical Sciences, Capital Normal University, Beijing 100048, China. Email: [gnge@zju.edu.cn](gnge@zju.edu.cn)  \nAbstract  \nThis paper investigates the existence, construction and classification of limited magnitude error-correcting codes, with a focus on splitter sets and their connections to group splittings. We establish new nonexistence results for quasi-perfect splitter sets and provide a complete classification of quasi-perfect B[0, 3](n) splitter sets in both singular and nonsingular cases. Furthermore, we derive improved lower bounds for the size of maximal B[0, 3](q) sets by investigating Cayley graphs, where q is a prime. We also provide existence criteria for perfect B[0, 6](q) splitter sets and quasi-perfect B [−4, 4](2p) sets for prime p. For perfect burst-correcting codes, we develop a general construction framework, and prove the existence of infinite families of (k2 , k1)-limitedmagnitude cyclic b-burst-correcting codes for k1 + k2 ≤ 4 and arbitrary burst length b. We further provide sufficient existence conditions for general parameters k1 and k2 . Our results combine algebraic, combinatorial, and number-theoretic methods to advance the understanding of codes tailored for flash memory and related storage systems.  \nIndex Terms  \nError correction codes, flash memory, lattice tilings, group factorization, group splitting.  \nI. INTRODUCTION  \nFlash memories are non-volatile, high density and low cost memories that have applications in many areas of modern life. For a higher density of flash memories, multilevel memory cells were introduced, which can store q levels, and common flash error mechanisms induce errors whose magnitudes (i.e., the number of level changes) are small. This setting stimulated research into specialized error-correcting codes for flash memory storage, called limited magnitude error correcting codes, first proposed in [4] and [14] . In this model, each symbol is an element in Z.1 Errors are modeled as bounded additive perturbations: for a codeword c = (c1 ,..., cn), a symbol ci may be distorted to ci + λ, where λ ∈ [−k1 , k2] = {−k1 , −k1 + 1 ,..., k2 } for some nonnegative integers k 1 and k2 with k2 > 0. The error is asymmetric if k 1 = 0; when k 1 = k2 , the error is symmetric. In general, such errors are called unbalanced as in [38] .  \nBesides flash memories, such codes also find wide applications in high-density magnetic recording channels [17], [18] and DNA-based storage systems [12], [35] .  \nTo give a formal definition for these codes, we first define the error ball  \nB (n, t, k2 , k 1 ) = 􀀚 e = (e1 , e2 ,..., en ) : eiwt∈(e)[ 1t, k2], 􀀛 ,  \nthat is, for every element of B (n, t, k2 , k 1 ), all its coordinates are zero, except for possibly at most t coordinates being in [−k1 , k2] \\ {0} =: [−k1 , k2]∗ . Our goal is to construct a code C ⊂ Zn such that for all x ∈ C, x + B(n, t, k2 , k 1 ) are disjoint. It is equivalent to saying that for all e ∈ B (n, t, k2 , k 1 ), e+C are disjoint. We refer to such codes as (k2 , k 1 )-limited magnitude t-error correcting codes, or (k2 , k 1 ) -limited magnitude single error correcting codes when t = 1 . If moreover, Zn = Fx∈C(x + B(n, t, k2 , k 1 )), we say that the code C is perfect.  \nLimited magnitude error correcting codes are apparently first investigated by Levenshtein and Vinck in [18] . What they called k-shift codes are codes cor","cbCaihAuRLjMVslm","https://ap.wps.com/l/cbCaihAuRLjMVslm","pdf",480264,2,1,28,"English","en",105,"# Introduction\n## Limited magnitude error model and notation\n## Splitter sets and perfect/quasi-perfect codes\n## Algebraic characterization via group splittings\n# Main contributions and constructions","[{\"question\":\"What problem does the paper address in limited magnitude error-correcting codes?\",\"answer\":\"It studies which limited magnitude codes exist, how to construct them, and how to classify them. The work focuses on splitter sets and their connection to group splittings.\"},{\"question\":\"How are flash memory errors modeled in the limited magnitude framework?\",\"answer\":\"Each symbol is perturbed by an additive amount λ within a bounded interval [−k1, k2], applied to at most t coordinates. The model becomes asymmetric when k1=0 and symmetric when k1=k2.\"},{\"question\":\"What results are obtained for splitter sets such as B[0, 3](n) and B[0, 6](q)?\",\"answer\":\"The paper proves new nonexistence results for quasi-perfect splitter sets and gives a complete classification of quasi-perfect B[0, 3](n) splitter sets for both singular and nonsingular cases. It also provides existence criteria for perfect B[0, 6](q) splitter sets.\"}]",1784198199,71,{"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},"new-results-on-limited-magnitude-error-correcting-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/new-results-on-limited-magnitude-error-correcting-codes/84782/",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 problem does the paper address in limited magnitude error-correcting codes?","Question",{"text":75,"@type":76},"It studies which limited magnitude codes exist, how to construct them, and how to classify them. The work focuses on splitter sets and their connection to group splittings.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are flash memory errors modeled in the limited magnitude framework?",{"text":80,"@type":76},"Each symbol is perturbed by an additive amount λ within a bounded interval [−k1, k2], applied to at most t coordinates. The model becomes asymmetric when k1=0 and symmetric when k1=k2.",{"name":82,"@type":73,"acceptedAnswer":83},"What results are obtained for splitter sets such as B[0, 3](n) and B[0, 6](q)?",{"text":84,"@type":76},"The paper proves new nonexistence results for quasi-perfect splitter sets and gives a complete classification of quasi-perfect B[0, 3](n) splitter sets for both singular and nonsingular cases. 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