{"id":"https://openalex.org/W4226066868","doi":"https://doi.org/10.1109/tit.2022.3170341","title":"Distribution of the Minimum Distance of Random Linear Codes","display_name":"Distribution of the Minimum Distance of Random Linear Codes","publication_year":2022,"publication_date":"2022-04-25","ids":{"openalex":"https://openalex.org/W4226066868","doi":"https://doi.org/10.1109/tit.2022.3170341"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2022.3170341","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3170341","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5087248118","display_name":"Jing Hao","orcid":null},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jing Hao","raw_affiliation_strings":["Department of Mathematics, Georgia Tech, Atlanta, GA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Georgia Tech, Atlanta, GA, USA","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101425204","display_name":"Han Huang","orcid":"https://orcid.org/0000-0001-5694-7701"},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Han Huang","raw_affiliation_strings":["Department of Mathematics, Georgia Tech, Atlanta, GA, USA"],"raw_orcid":"https://orcid.org/0000-0001-5694-7701","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Georgia Tech, Atlanta, GA, USA","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075539741","display_name":"Galyna V. Livshyts","orcid":null},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Galyna V. Livshyts","raw_affiliation_strings":["Department of Mathematics, Georgia Tech, Atlanta, GA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Georgia Tech, Atlanta, GA, USA","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5037104598","display_name":"Konstantin Tikhomirov","orcid":null},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Konstantin Tikhomirov","raw_affiliation_strings":["Department of Mathematics, Georgia Tech, Atlanta, GA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Georgia Tech, Atlanta, GA, USA","institution_ids":["https://openalex.org/I130701444"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I130701444"],"apc_list":null,"apc_paid":null,"fwci":0.1312,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.49880792,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"68","issue":"10","first_page":"6388","last_page":"6401"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10796","display_name":"Cooperative Communication and Network Coding","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11321","display_name":"Error Correcting Code Techniques","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5435022115707397},{"id":"https://openalex.org/keywords/hamming-distance","display_name":"Hamming distance","score":0.5420324802398682},{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.5356573462486267},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.4519370496273041},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.448917418718338},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4298977255821228},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.42528772354125977},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.1160973608493805},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09064742922782898}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5435022115707397},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.5420324802398682},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.5356573462486267},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.4519370496273041},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.448917418718338},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4298977255821228},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.42528772354125977},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.1160973608493805},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09064742922782898}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2022.3170341","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3170341","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8922158340","display_name":null,"funder_award_id":"CAREER DMS-1753260","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W574013035","https://openalex.org/W1911679041","https://openalex.org/W1981078592","https://openalex.org/W2003177806","https://openalex.org/W2014860393","https://openalex.org/W2014944833","https://openalex.org/W2034215849","https://openalex.org/W2049724793","https://openalex.org/W2096667287","https://openalex.org/W2109084006","https://openalex.org/W2126377499","https://openalex.org/W2128765501","https://openalex.org/W2129496126","https://openalex.org/W2162185689","https://openalex.org/W2164530213","https://openalex.org/W2175843853","https://openalex.org/W2886233444","https://openalex.org/W4234145994","https://openalex.org/W4313550028"],"related_works":["https://openalex.org/W2504004674","https://openalex.org/W2498744856","https://openalex.org/W1595229445","https://openalex.org/W4390482104","https://openalex.org/W322408318","https://openalex.org/W149041114","https://openalex.org/W2963177394","https://openalex.org/W1965815883","https://openalex.org/W4313359513","https://openalex.org/W763418848"],"abstract_inverted_index":{"Let":[0],"<inline-formula":[1,34,41,69,85,97,105,125,194],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[2,35,42,66,70,86,98,106,126,195],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">":[3,36,43,71,87,99,107,127,196],"<tex-math":[4,37,44,72,88,100,108,128,197],"notation=\"LaTeX\">$q\\geq":[5],"2$":[6],"</tex-math></inline-formula>":[7,39,47,74,90,103,110,131,201],"be":[8],"a":[9,28,118,134,144,170,179],"prime":[10],"power.":[11],"In":[12],"this":[13],"paper,":[14],"we":[15,185],"study":[16],"the":[17,20,24,55,59,77,82,152,157,162,166,174,182,190],"distribution":[18,56,79,120],"of":[19,27,32,58,81,84,151,154,165,181,189],"minimum":[21,60,83,175],"distance":[22,61],"(in":[23],"Hamming":[25],"metric)":[26],"random":[29,93,167],"linear":[30,163],"code":[31,176],"dimension":[33],"notation=\"LaTeX\">$k$":[38],"in":[40,68,114,137,143],"notation=\"LaTeX\">$\\mathbb":[45],"{F}_{q}^{n}$":[46],".":[48,111,202],"We":[49],"provide":[50],"quantitative":[51],"estimates":[52],"showing":[53],"that":[54,147],"function":[57,80],"is":[62],"close":[63],"(":[64],"<italic":[65],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">superpolynomially</i>":[67],"notation=\"LaTeX\">$n$":[73,109],")":[75],"to":[76,117,133],"cumulative":[78],"notation=\"LaTeX\">$(q^{k}-1)/(q-1)$":[89],"independent":[91],"binomial":[92],"variables":[94],"with":[95],"parameters":[96],"notation=\"LaTeX\">$\\frac":[101,129],"{1}{q}$":[102],"and":[104],"The":[112],"latter,":[113],"turn,":[115],"converges":[116,132],"Gumbel":[119],"at":[121],"integer":[122],"points":[123],"when":[124],"{k}{n}$":[130],"fixed":[135],"number":[136],"(0,":[138],"1).":[139],"Our":[140],"result":[141],"confirms":[142],"strong":[145],"sense":[146],"apart":[148],"from":[149],"identification":[150],"weights":[153],"proportional":[155],"codewords,":[156],"probabilistic":[158],"dependencies":[159],"introduced":[160],"by":[161],"structure":[164],"code,":[168],"produce":[169],"negligible":[171],"effect":[172],"on":[173],"weight.":[177],"As":[178],"corollary":[180],"main":[183],"result,":[184],"obtain":[186],"an":[187],"improvement":[188],"Gilbert\u2013Varshamov":[191],"bound":[192],"for":[193],"notation=\"LaTeX\">$2&lt;":[198],"q&lt;":[199],"49$":[200]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
