[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86496-en":3,"doc-seo-86496-105":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},86496,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","List-Decoding Counterexamples Yield Lower Bounds on Mutual Correlated Agreement Error","Mutual correlated agreement measures whether random linear combinations of received words can create a new, unexpectedly large agreement with a code—an effect crucial to batched proximity testing soundness. The work constructively shows that list-decoding counterexamples imply lower bounds on mutual correlated agreement error. Starting from explicit (p,L)-list-decodability counterexamples, it builds a code C′ with controlled minimum-distance loss and explicit witnessing word pairs. It further extends the construction structure-preservingly to coordinate-indexed families and applies it to AG evaluation codes and Reed–Solomon codes.","arXiv :2607 . 10572v 1 [ cs .IT] 12 Jul 2026  \nList-Decoding Counterexamples Yield Lower Bounds on Mutual Correlated Agreement Error  \nYiwen Gao Hong Yang Yang Xu Haibin Kan∗  \nJuly 14, 2026  \nAbstract  \nMutual correlated agreement captures whether a random linear combination of received words can create a new large agreement with a code, a property relevant to the soundness of batched proximity testing. We show constructively that list-decoding counterexamples yield lower bounds on the mutual correlated agreement error. Given an explicit counterexample to the (p, L)-list-decodability of a linear code over Fq , we construct a related code C′ of the same length and dimension such that errMCA (C′, p) ≥ ~~1~~q l ~~ ~~(Lq1L)qm, while decreasing its minimum distance by at most one. The construction also produces an explicit pair of words witnessing this error.  \nWe further give a structure-preserving version for code families whose coordinates are indexed by a finite set Ω, with each index determining a generator-matrix column through a map v : Ω → Fkq . The construction changes at most one coordinate index and ensures that the output code remains in the same indexed family. As applications, we instantiate this principle for algebraic-geometry (AG) evaluation codes and Reed– Solomon codes. For AG codes, if G is the divisor defining the underlying Riemann–Roch space and N is the number of rational places outside supp(G) available for evaluation, the resulting code remains over the same function field and Riemann–Roch space, with a modified set of evaluation places. Its mutual correlated agreement error is at least ~~1~~q l N(+1)NL~~ ~~degGm. The Reed–Solomon conclusion follows as the Vandermonde-column specialization.  \n∗ Yiwen Gao, Hong Yang, Yang Xu, and Haibin Kan are with the College of Computer Science and Artificial Intelligence, Fudan University, Shanghai 200433, China, and Shanghai Engineering Research Center of Blockchain, Shanghai 200433, China. Haibin Kan is also with Shanghai Institute for Mathematics and Interdisciplinary Sciences, Shanghai 200433,  \nChina. Emails: [ywgao21@m.fudan.edu.cn](ywgao21@m.fudan.edu.cn), [hongyang1358@gmail.com](hongyang1358@gmail.com), [xuyyang@fudan.edu.cn](xuyyang@fudan.edu.cn),  \n[hbkan@fudan.edu.cn](hbkan@fudan.edu.cn).  \n1 Introduction  \nLinear codes, defined as linear subspaces C ⊆ Fnq, play a vital role in modern digital communication and cryptography due to their elegant algebraic structure and efficient encoding mechanisms. In particular, they have emerged asa cornerstone in the design of succinct non-interactive arguments of knowledge (SNARKs) . A critical component enabling these efficient proof systems is the proximity test, a probabilistic algorithm that allows a verifier to efficiently determine whether a queried vector is close to a valid codeword by examining only a small number of its coordinates. In practice, it is often necessary to perform proximity tests on a large batch of vectors. To reduce verification overhead, a common approach is to randomly combine these vectors into a single vector and apply the proximity test exclusively to this linear combination. The soundness of such testing mechanisms inherently relies on the distance-preserving properties of the underlying codes.  \nIntuitively, the distance-preserving capability of a code guarantees that if a set of words is far from the code, a random linear combination of these words will also remain far from the code with high probability. This principle is central to the soundness analysis of IOPPs and code-based proof systems, and variants of this property have been studied in many works [RVW13; Ame+17; Ben+18; Ben+20] . To formally capture and quantify this behavior, several coding-theoretic notions have been introduced, including proximity gaps, correlated agreement, and mutual correlated agreement; see, e.g.,[Ben+23; Arn+25] . Mutual correlated agreement gives a more refined account of this phenomenon: it requires th","cbCaitQhMsjaC9Ij","https://ap.wps.com/l/cbCaitQhMsjaC9Ij","pdf",461989,4,1,24,"English","en",105,"# Abstract\n# Introduction\n## Background: proximity testing and correlated agreement\n## Prior work: positive results and limitations\n## Paper contributions and construction overview","[{\"question\":\"What does mutual correlated agreement error quantify in batched proximity testing?\",\"answer\":\"It quantifies the likelihood that a random linear combination of received words creates a large agreement domain with the code that is not already explained by the correlated agreement structure of the original words.\"},{\"question\":\"How do list-decoding counterexamples lead to lower bounds on mutual correlated agreement error?\",\"answer\":\"Given an explicit counterexample to (p,L)-list-decodability of a linear code over Fq, the paper constructs a related code C′ of the same length and dimension and proves that its mutual correlated agreement error is bounded below, while decreasing minimum distance by at most one.\"},{\"question\":\"What role does the structure-preserving construction play, and where is it applied?\",\"answer\":\"It adapts the construction to code families indexed by a finite set Ω via a mapping from indices to generator-matrix columns, changing at most one coordinate index while keeping the code in the same indexed family. Applications instantiate the principle for algebraic-geometry evaluation codes and Reed–Solomon codes.\"}]",1784212177,60,{"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},"list-decoding-counterexamples-yield-lower-bounds-on-mutual-correlated-agreement-error","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/list-decoding-counterexamples-yield-lower-bounds-on-mutual-correlated-agreement-error/86496/",{"url":52,"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","2026-07-27","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 does mutual correlated agreement error quantify in batched proximity testing?","Question",{"text":75,"@type":76},"It quantifies the likelihood that a random linear combination of received words creates a large agreement domain with the code that is not already explained by the correlated agreement structure of the original words.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do list-decoding counterexamples lead to lower bounds on mutual correlated agreement error?",{"text":80,"@type":76},"Given an explicit counterexample to (p,L)-list-decodability of a linear code over Fq, the paper constructs a related code C′ of the same length and dimension and proves that its mutual correlated agreement error is bounded below, while decreasing minimum distance by at most one.",{"name":82,"@type":73,"acceptedAnswer":83},"What role does the structure-preserving construction play, and where is it applied?",{"text":84,"@type":76},"It adapts the construction to code families indexed by a finite set Ω via a mapping from indices to generator-matrix columns, changing at most one coordinate index while keeping the code in the same indexed family. Applications instantiate the principle for algebraic-geometry evaluation codes and Reed–Solomon codes.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,109,114,119,122,127,130,134],{"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":20,"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":29,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"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":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]