{"id":"https://openalex.org/W4379987773","doi":"https://doi.org/10.1017/s0963548323000111","title":"On the number of error correcting codes","display_name":"On the number of error correcting codes","publication_year":2023,"publication_date":"2023-06-09","ids":{"openalex":"https://openalex.org/W4379987773","doi":"https://doi.org/10.1017/s0963548323000111"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548323000111","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548323000111","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","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/A5054551001","display_name":"Dingding Dong","orcid":"https://orcid.org/0000-0001-8500-2897"},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Dingding Dong","raw_affiliation_strings":["Department of Mathematics, Harvard University, Cambridge, MA 02138, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Harvard University, Cambridge, MA 02138, USA","institution_ids":["https://openalex.org/I136199984"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037481362","display_name":"Nitya Mani","orcid":"https://orcid.org/0000-0003-0348-5886"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Nitya Mani","raw_affiliation_strings":["Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5021133458","display_name":"Yufei Zhao","orcid":"https://orcid.org/0000-0002-1995-3755"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yufei Zhao","raw_affiliation_strings":["Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5037481362"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":0.3497,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.63542171,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"32","issue":"5","first_page":"819","last_page":"832"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9998000264167786,"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":0.9998000264167786,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7966099977493286},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6929175853729248},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.6541965007781982},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.6434857249259949},{"id":"https://openalex.org/keywords/hamming-code","display_name":"Hamming code","score":0.6343121528625488},{"id":"https://openalex.org/keywords/hamming-distance","display_name":"Hamming distance","score":0.6238552331924438},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.45053809881210327},{"id":"https://openalex.org/keywords/ball","display_name":"Ball (mathematics)","score":0.4315633773803711},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4213100075721741},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.12308245897293091},{"id":"https://openalex.org/keywords/block-code","display_name":"Block code","score":0.12155625224113464},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07617759704589844},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07250109314918518},{"id":"https://openalex.org/keywords/decoding-methods","display_name":"Decoding methods","score":0.04937797784805298},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.041548311710357666}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7966099977493286},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6929175853729248},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.6541965007781982},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6434857249259949},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.6343121528625488},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.6238552331924438},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.45053809881210327},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.4315633773803711},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4213100075721741},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.12308245897293091},{"id":"https://openalex.org/C157125643","wikidata":"https://www.wikidata.org/wiki/Q884707","display_name":"Block code","level":3,"score":0.12155625224113464},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07617759704589844},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07250109314918518},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.04937797784805298},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.041548311710357666},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1017/s0963548323000111","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548323000111","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W109405981","https://openalex.org/W1555293549","https://openalex.org/W1597478441","https://openalex.org/W2035763891","https://openalex.org/W2109084006","https://openalex.org/W2963239130","https://openalex.org/W2963669358","https://openalex.org/W2963846904","https://openalex.org/W4301288248","https://openalex.org/W6604469142","https://openalex.org/W6706667872"],"related_works":["https://openalex.org/W1503555387","https://openalex.org/W2115422885","https://openalex.org/W2258594450","https://openalex.org/W4235874751","https://openalex.org/W2169207476","https://openalex.org/W4320722052","https://openalex.org/W2618304594","https://openalex.org/W4298743697","https://openalex.org/W2065738971","https://openalex.org/W2739495308"],"abstract_inverted_index":{"Abstract":[0],"We":[1],"show":[2],"that":[3],"for":[4,28,74],"a":[5,63],"fixed":[6],"$q$":[7,12],",":[8,39],"the":[9,47,53,56,72],"number":[10],"of":[11,18,55,65],"-ary":[13],"$t$":[14,58],"-error":[15],"correcting":[16],"codes":[17],"length":[19],"$n$":[20],"is":[21,46,52],"at":[22],"most":[23],"$2^{(1":[24],"+":[25],"o(1))":[26],"H_q(n,t)}$":[27],"all":[29],"$t":[30,75],"\\leq":[31],"(1":[32],"-":[33,35],"q^{-1})n":[34],"2\\sqrt{n":[36],"\\log":[37],"n}$":[38],"where":[40],"$H_q(n,":[41],"t)":[42],"=":[43,76],"q^n/":[44],"V_q(n,t)$":[45],"Hamming":[48,59],"bound":[49],"and":[50,68],"$V_q(n,t)$":[51],"cardinality":[54],"radius":[57],"ball.":[60],"This":[61],"proves":[62],"conjecture":[64],"Balogh,":[66],"Treglown,":[67],"Wagner,":[69],"who":[70],"showed":[71],"result":[73],"o(n^{1/3}":[77],"(\\log":[78],"n)^{-2/3})$":[79],".":[80]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-01-24T23:23:39.755997","created_date":"2025-10-10T00:00:00"}
