[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-638607-105":59,"doc-detail-638607-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","codes-from-polynomials-over-finite-fields-jmm-2021","Codes from Polynomials over Finite Fields - JMM 2021","","Coding Theory lecture notes introducing reliable communication over noisy channels through redundancy using finite-field codes. The material starts from a binary transmission example, then defines linear codes over Fq, including Hamming distance and weight. It explains minimum distance, error-correction capability, and the central combinatorial problem of maximizing code size given constraints on minimum distance. It also highlights Reed–Solomon codes as a key topic and provides example bounds and values such as A2(17;8)=36.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/codes-from-polynomials-over-finite-fields-jmm-2021/638607/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/codes-from-polynomials-over-finite-fields-jmm-2021/638607.png","ImageObject",300,407,{"name":92,"@type":93},"Bulrr","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-10-22",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How does redundancy improve communication over a noisy channel?","Question",{"text":112,"@type":113},"Instead of sending a single bit, the idea is to send a repeated block (e.g., 000 or 111). The receiver decodes to the codeword that is more likely given the observed errors, increasing the probability of correct recovery.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What are the key definitions for codes over finite fields?",{"text":117,"@type":113},"The notes define a code C of length n as a subset of Fnq, and a linear code as a linear subspace. It then introduces Hamming distance between vectors and Hamming weight as the distance to the zero vector.",{"name":119,"@type":110,"acceptedAnswer":120},"How do minimum distance and error correction relate?",{"text":121,"@type":113},"The minimum distance d(C) is defined as the smallest distance between distinct codewords. For a code with minimum distance d, it can correct up to t = floor((d-1)/2) errors, and for linear codes this corresponds to the minimum weight among nonzero codewords.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},638607,1793042763,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},5909892115043,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Codes from Polynomials over Finite Fields  \nNathan Kaplan  \nUniversity of California, Irvine  \nJoint Mathematics Meetings 2021  \nJanuary 7, 2021  \nKaplan (UCI) Codes from Polynomials January 7, 2021 1 / 33  \nI. What is Coding Theory All About?  \nKaplan (UCI) Codes from Polynomials January 7, 2021 2 / 33  \nSuppose we want to communicate over a noisy channel. I will send you a message: 0 or a 1 .  \n If I send 0, there is a 90% chance you receive 0 .  \n If I send 1, there is a 90% chance you receive 1 . Idea: Instead of sending 0 or 1, I will send 000 or 111 .  \n􀀎 If you receive 010, you `decode' as 000 because it is likelier that I sent 000 and that there was 1 error than it is that I sent 111 and there were 2 errors.  \n If I send 0 or 1, there is a 90% chance you receive the correct message.  If I send 000 or 111, you receive the correct message with probability  \n(:9)3 + 􀀒 31􀀓 (:9)2 (:1) = :972:  \nThere is a cost for this increased reliability– have to send 3 bits instead of 1 .  \nKaplan (UCI) Codes from Polynomials January 7, 2021 3 / 33  \nMajor Question  \nHow do we efficiently build redundancy into our set of messages so that we can identify and correct errors?  \nKaplan (UCI) Codes from Polynomials January 7, 2021 4 / 33  \nCoding Theory Basics I  \nLet Fq be a finite field of size q.  \nDefinition  \n A code over Fq of length n is a subset C 􀀒 Fnq .  \n C is a linear code if it is a linear subspace ofFnq .  \nThat is, if c 1 ; c2 2 C then c 1 + c2 2 C and 􀀋c 1 2 C for any 􀀋 2 Fq .  For xy((x1y1 ;;:::::;:;xnyn)) 2 Fnq, the Hamming distance between x and y is  \nd (x ; y ) = \\#fi j xi  yi g:  \n The Hamming weight of x is wt (x) = d (x ; 0) = \\#fi j xi  0g.  \nf (0; 0 ; 0); (1; 1 ; 1)g 􀀚 F32 is a 1-dimensional linear code.  \nd((0; 0 ; 0); (1; 1 ; 1)) = 3 :  \nKaplan (UCI) Codes from Polynomials January 7, 2021 5 / 33  \nCoding Theory Basics II  \nDefinition  \nThe minimum distance of a code C is  \nd (C ) = min d (x ; y): x ;y 2C  \nxy  \n􀀎 If C is linear, d (C ) is the minimum weight of a nonzero c 2 C.  \nd (x ; y) = d (x 􀀀 y ; y 􀀀 y) = wt (x 􀀀 y)  \n􀀎 In a code with minimum distance d, can correct up to t = b~~d ~~~~1~~ c errors.  \nC = f (0; 0 ; 0); (1; 1 ; 1)g 􀀚 F32 has d (C ) = 3.  \nYou can correct up to t = b 31 c = 1 error.  \nKaplan (UCI) Codes from Polynomials January 7, 2021 6 / 33  \nMain Problem in Combinatorial Coding Theory  \nWe want codes C 􀀒 Fnq of large size and large minimum distance.  \nDefinition  \nLet Aq(n ; d) be the maximum size of a code C 􀀒 Fnq that has minimum  \ndistance at least d.   \nMain Problem in Combinatorial Coding Theory: Compute values of Aq(n ; d) .  \nKaplan (UCI) Codes from Polynomials January 7, 2021 7 / 33  \n[Östergård, 2011] : A2 (17; 8) = 36.  \nFigure: Brouwer's tables of upper and lower bounds for A2 (n ; d)  \nWhat is A2 (17; 6)?  \nKaplan (UCI) Codes from Polynomials January 7, 2021 8 / 33  \nKaplan (UCI) Codes from Polynomials January 7, 2021 9 / 33  \nII. Reed-Solomon Codes  \nKaplan (UCI) Codes from Polynomials January 7, 2021 10 / 33","cbCaihR9j1LCr6sV","https://ap.wps.com/l/cbCaihR9j1LCr6sV","pdf",1073453,33,"English","# I. What is Coding Theory All About?\n## Coding Theory Basics I\n## Coding Theory Basics II\n# Main Problem in Combinatorial Coding Theory\n# II. Reed-Solomon Codes","[{\"question\":\"How does redundancy improve communication over a noisy channel?\",\"answer\":\"Instead of sending a single bit, the idea is to send a repeated block (e.g., 000 or 111). The receiver decodes to the codeword that is more likely given the observed errors, increasing the probability of correct recovery.\"},{\"question\":\"What are the key definitions for codes over finite fields?\",\"answer\":\"The notes define a code C of length n as a subset of Fnq, and a linear code as a linear subspace. It then introduces Hamming distance between vectors and Hamming weight as the distance to the zero vector.\"},{\"question\":\"How do minimum distance and error correction relate?\",\"answer\":\"The minimum distance d(C) is defined as the smallest distance between distinct codewords. For a code with minimum distance d, it can correct up to t = floor((d-1)/2) errors, and for linear codes this corresponds to the minimum weight among nonzero codewords.\"}]","Codes from Polynomials over Finite Fields - JMM 2021 | PDF",1792689665,83]