[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81561-en":3,"doc-seo-81561-105":29,"detail-sidebar-cat-0-en-105":82},{"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":4,"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},81561,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","List Decoding of Reed Solomon Codes and Folded Reed Solomon Codes Over Galois Ring","List decoding generalizes unique decoding by outputting all codewords within a prescribed Hamming radius when the error count exceeds half the minimum distance. While list decoding has been intensively studied over finite fields, extending proximity-gap and decoding guarantees to Galois rings is challenging. This work extends the Guruswami–Sudan procedure to Reed–Solomon codes over Galois rings, achieving list decoding radius 1−√r. It further analyzes folded Reed–Solomon codes, reaching the Singleton bound analogue, and develops pruning and list-size bounds with both algorithmic and combinatorial guarantees.","arXiv :2511 .04 135v 3 [ cs .IT] 10 Jul 2026  \nList Decoding of Reed–Solomon Codes and Folded Reed–Solomon Codes Over Galois Ring ∗  \nChen Yuan, Ruiqi Zhu  \nAbstract  \nList decoding of codes can be seen as the generalization of unique decoding of codes. While list decoding over finite fields has been extensively studied, extending these results to more general algebraic structures such as Galois rings remains an important challenge. Due to recent progress in zero knowledge systems, there is a growing demand to investigate the proximity gap of codes over Galois rings [JLX + 25, GLS+ 23, WZD25] . The proximity gap is closely related to the decoding capability of codes. It was shown [BCI+ 20] that the proximity gap for RS codes over finite field can be improved to 1 − √r if one consider list decoding instead of unique decoding. However, we know very little about RS codes over Galois ring which might hinder the development of zero knowledge proof system for ring-based arithmetic circuit. In this work, we first extend the list decoding procedure of Guruswami and Sudan to Reed-Solomon codes over Galois rings, which shows that RS codes with rate r can be list decoded up to radius 1 − √r. Then, we investigate the list decoding of folded Reed-Solomon codes over Galois rings. We show that the list decoding radius of folded Reed-Solomon codes can reach the Singlton bound as its counterpart over finite field. We also extend the deterministic pruning method of [AHS26] to Galois rings, showing how to prune the affine free module obtained from the linear-algebraic decoder and recover the candidate codewords. Finally, we obtain an algorithmic list-size bound of O(1/ε2 ) for our folded Reed–Solomon code by extending the approach of [Sri25] to Galois rings. Moreover, at the combinatorial level, by developing the recent work of [CZ25], we show that folded Reed–Solomon codes over Galois rings satisfy the relaxed generalized Singleton bound in the average-radius sense with optimal list size O(1/ε) . Specifically, this result is not obtained by a straightforward extension of [CZ25] from finite fields to Galois rings; instead, we develop a more delicate tree-based argument that exploits the p-adic congruence structure of Galois rings.  \n∗ C. Yuan is with School of Computer Science, Shanghai Jiao Tong University. (Email: [chen_yuan@sjtu.edu.cn](chen_yuan@sjtu.edu.cn)) R. Zhu is with School of Computer Science, Shanghai Jiao Tong University.(Email: [sjtuzrq7777@sjtu.edu.cn](sjtuzrq7777@sjtu.edu.cn))  \n1 Introduction  \nList decoding, first introduced in [Eli57], provides a way to recover codewords even when the number of errors e goes beyond half of the minimum distance d. Specifically, if the number of errors e in a received word exceeds ⌊ (d − 1)/2⌋, it is possible that more than one codeword that is within (Hamming) distance e from the received word. In this case, a list decoder outputs all codewords that fall within this Hamming ball of radius e.  \nReed-Solomon codes (RS codes for short), were first proposed in 1960 [RS60] . RS codes belong to a family of Since RS codes belong to the family of the maximum distance separable (MDS) codes. RS codes also have very efficient encoding and decoding algorithms [Ber15] and [SKHN75] . Let ρ be the decoding radius and R be the rate of a code. Sudan [Sud97] introduced the first explicit list decoding algorithm for RS codes that can decoded RS codes beyond unique decoding radius. Subsequently, Guruswami and Sudan [GS98] refined that algorithm to achieve Johnson bound for any rate. Furthermore, their method can also be extended to the decoding of algebraic geometry codes which initiated an intensive line of research that produced numerous results in the field of list decoding [KV03, PW04, RR02, TR03] . Understanding the limits of list-decoding and listrecovery of RS codes is of prime interest in coding theory and has attracted a lot of attention over the past decades. In a recent breakthrough, Shangguan and Tamo","cbCaihYFXiVnRGPq","https://ap.wps.com/l/cbCaihYFXiVnRGPq","pdf",548461,1,45,"English","en",105,"# Introduction\n## Background on list decoding\n## Reed–Solomon and folded Reed–Solomon codes\n## Previous results and motivation","[{\"question\":\"What additional methods and bounds are developed beyond decoding radius?\",\"answer\":\"The document extends deterministic pruning to Galois rings, recovers candidate codewords from the linear-algebraic decoder output, and provides algorithmic list-size bounds for folded Reed–Solomon codes along with a combinatorial relaxed generalized Singleton bound in the average-radius sense.\"}]",1784174330,113,{"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":77,"head_meta":79,"extra_data":81,"updated_unix":27},"list-decoding-of-reed-solomon-codes-and-folded-reed-solomon-codes-over-galois-ring","",{"@graph":35,"@context":76},[36,53,67],{"@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/list-decoding-of-reed-solomon-codes-and-folded-reed-solomon-codes-over-galois-ring/81561/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70],{"name":71,"@type":72,"acceptedAnswer":73},"What additional methods and bounds are developed beyond decoding radius?","Question",{"text":74,"@type":75},"The document extends deterministic pruning to Galois rings, recovers candidate codewords from the linear-algebraic decoder output, and provides algorithmic list-size bounds for folded Reed–Solomon codes along with a combinatorial relaxed generalized Singleton bound in the average-radius sense.","Answer","https://schema.org",{"og:url":51,"og:type":78,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":80,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":83},[84,88,92,96,101,106,111,114,119,122,126],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":45,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":45,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":112,"slug":113},30,"research-report",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":45,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":97,"slug":129},19,"General","general"]