{"id":"https://openalex.org/W4406891880","doi":"https://doi.org/10.1109/tit.2025.3535700","title":"The Sufficient and Necessary Conditions for the Minimum Distance of the BCH Code <i>C</i>(q,q+1,3,h) to Be 3 and 4","display_name":"The Sufficient and Necessary Conditions for the Minimum Distance of the BCH Code <i>C</i>(q,q+1,3,h) to Be 3 and 4","publication_year":2025,"publication_date":"2025-01-28","ids":{"openalex":"https://openalex.org/W4406891880","doi":"https://doi.org/10.1109/tit.2025.3535700"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2025.3535700","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2025.3535700","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/A5107912814","display_name":"H. Xu","orcid":"https://orcid.org/0000-0002-0714-8628"},"institutions":[{"id":"https://openalex.org/I76569877","display_name":"Southeast University","ror":"https://ror.org/04ct4d772","country_code":"CN","type":"education","lineage":["https://openalex.org/I76569877"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Haojie Xu","raw_affiliation_strings":["School of Mathematics, Southeast University, Nanjing, China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, Southeast University, Nanjing, China","institution_ids":["https://openalex.org/I76569877"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101436835","display_name":"Xia Wu","orcid":"https://orcid.org/0000-0001-8640-6156"},"institutions":[{"id":"https://openalex.org/I76569877","display_name":"Southeast University","ror":"https://ror.org/04ct4d772","country_code":"CN","type":"education","lineage":["https://openalex.org/I76569877"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xia Wu","raw_affiliation_strings":["School of Mathematics, Southeast University, Nanjing, China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, Southeast University, Nanjing, China","institution_ids":["https://openalex.org/I76569877"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Wei Lu","orcid":"https://orcid.org/0009-0005-5826-488X"},"institutions":[{"id":"https://openalex.org/I76569877","display_name":"Southeast University","ror":"https://ror.org/04ct4d772","country_code":"CN","type":"education","lineage":["https://openalex.org/I76569877"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wei Lu","raw_affiliation_strings":["School of Mathematics, Southeast University, Nanjing, China"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, Southeast University, Nanjing, China","institution_ids":["https://openalex.org/I76569877"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5026635882","display_name":"Xiwang Cao","orcid":"https://orcid.org/0000-0002-6950-8588"},"institutions":[{"id":"https://openalex.org/I9842412","display_name":"Nanjing University of Aeronautics and Astronautics","ror":"https://ror.org/01scyh794","country_code":"CN","type":"education","lineage":["https://openalex.org/I9842412"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiwang Cao","raw_affiliation_strings":["Department of Math, Nanjing University of Aeronautics and Astronautics, Nanjing, China"],"affiliations":[{"raw_affiliation_string":"Department of Math, Nanjing University of Aeronautics and Astronautics, Nanjing, China","institution_ids":["https://openalex.org/I9842412"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5107912814"],"corresponding_institution_ids":["https://openalex.org/I76569877"],"apc_list":null,"apc_paid":null,"fwci":7.0705,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.95928583,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":98},"biblio":{"volume":"71","issue":"6","first_page":"4206","last_page":"4213"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9736999869346619,"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":0.9736999869346619,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/bch-code","display_name":"BCH code","score":0.9473167061805725},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4824860394001007},{"id":"https://openalex.org/keywords/code","display_name":"Code (set theory)","score":0.4700663983821869},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.44352322816848755},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.4044608175754547},{"id":"https://openalex.org/keywords/decoding-methods","display_name":"Decoding methods","score":0.37931734323501587},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.36496850848197937},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.32143962383270264},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.2049776017665863},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.12700054049491882}],"concepts":[{"id":"https://openalex.org/C42276685","wikidata":"https://www.wikidata.org/wiki/Q795705","display_name":"BCH code","level":3,"score":0.9473167061805725},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4824860394001007},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.4700663983821869},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44352322816848755},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.4044608175754547},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.37931734323501587},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.36496850848197937},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.32143962383270264},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.2049776017665863},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.12700054049491882},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2025.3535700","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2025.3535700","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":[{"score":0.5,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[{"id":"https://openalex.org/G2786323647","display_name":null,"funder_award_id":"12171241","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6589418543","display_name":null,"funder_award_id":"12371035","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7923415795","display_name":null,"funder_award_id":"BK20233002","funder_id":"https://openalex.org/F4320329290","funder_display_name":"China Ship Scientific Research Center"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320329290","display_name":"China Ship Scientific Research Center","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W584522352","https://openalex.org/W1530125006","https://openalex.org/W1957362188","https://openalex.org/W1993830711","https://openalex.org/W2020484625","https://openalex.org/W2034347399","https://openalex.org/W2162775065","https://openalex.org/W2901522846","https://openalex.org/W3022561029","https://openalex.org/W3094527023","https://openalex.org/W4210408507","https://openalex.org/W4239704585","https://openalex.org/W4289821394","https://openalex.org/W4292937362","https://openalex.org/W4311625003","https://openalex.org/W4315490130","https://openalex.org/W4320016034","https://openalex.org/W4322725230","https://openalex.org/W4377263704","https://openalex.org/W4379623567","https://openalex.org/W4388921248","https://openalex.org/W4389665371"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2912502034","https://openalex.org/W1995527419","https://openalex.org/W1976834495","https://openalex.org/W1768316021","https://openalex.org/W4242597560","https://openalex.org/W3197394982","https://openalex.org/W1968289971","https://openalex.org/W4300588357","https://openalex.org/W2129090975"],"abstract_inverted_index":{"In":[0],"this":[1,109],"paper,":[2],"the":[3,9,13,34,38,101,106],"sufficient":[4,139],"and":[5,26,64,86,149],"necessary":[6],"conditions":[7,99],"for":[8,54,76,132,141],"minimum":[10,35],"distance":[11,36,126],"of":[12,37,103,108,123],"BCH":[14,39,110],"code":[15,40,111],"<inline-formula":[16,41,57,67,79,89],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[17,42,58,68,80,90],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">":[18,43,59,69,81,91],"<tex-math":[19,44,60,70,82,92],"notation=\"LaTeX\">$\\mathcal":[20,45],"{C}_{(q,q+1,3,h)}$":[21,46],"</tex-math></inline-formula>":[22,62,84],"to":[23,146],"be":[24,33,147],"3":[25],"4":[27],"are":[28,51,112,135,162],"provided,":[29],"respectively.":[30],"Let":[31],"d":[32],"</tex-math></inline-formula>.":[47,95],"The":[48],"following":[49],"results":[50],"proved:":[52],"1)":[53],"any":[55],"q,":[56],"notation=\"LaTeX\">$d=3$":[61],"if":[63,66,85,88],"only":[65,87],"notation=\"LaTeX\">$\\gcd":[71,93],"(2h+1,q+1)\\gt":[72],"1$":[73],"</tex-math></inline-formula>;":[74],"2)":[75],"q":[77,116],"odd,":[78],"notation=\"LaTeX\">$d=4$":[83],"(2h+1,q+1)=1$":[94],"By":[96],"combining":[97],"these":[98,104,142,158],"with":[100],"dimensions":[102],"codes,":[105],"parameters":[107],"determined":[113],"completely":[114],"when":[115],"is":[117,154],"odd.":[118],"Moreover,":[119],"several":[120,160],"infinite":[121],"families":[122],"almost":[124,143],"maximum":[125],"separable":[127],"(almost":[128],"MDS":[129,144],"or":[130],"AMDS":[131],"short)":[133],"codes":[134,145,153],"derived.":[136],"Furthermore,":[137],"a":[138],"condition":[140],"distance-optimal":[148],"dimension-optimal":[150],"locally":[151],"repairable":[152],"presented.":[155],"Based":[156],"on":[157],"conditions,":[159],"examples":[161],"also":[163],"given.":[164]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-01-29T00:00:00"}