{"id":"https://openalex.org/W7160635463","doi":"https://doi.org/10.1016/j.ffa.2026.102844","title":"The extended codes of NMDS codes","display_name":"The extended codes of NMDS codes","publication_year":2026,"publication_date":"2026-05-08","ids":{"openalex":"https://openalex.org/W7160635463","doi":"https://doi.org/10.1016/j.ffa.2026.102844"},"language":"en","primary_location":{"id":"doi:10.1016/j.ffa.2026.102844","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.ffa.2026.102844","pdf_url":null,"source":{"id":"https://openalex.org/S925225069","display_name":"Finite Fields and Their Applications","issn_l":"1071-5797","issn":["1071-5797","1090-2465"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Finite Fields and Their Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1016/j.ffa.2026.102844","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5086111868","display_name":"Qian Ma","orcid":"https://orcid.org/0000-0001-6398-4949"},"institutions":[{"id":"https://openalex.org/I16365422","display_name":"Hefei University of Technology","ror":"https://ror.org/02czkny70","country_code":"CN","type":"education","lineage":["https://openalex.org/I16365422"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Qian Ma","raw_affiliation_strings":["School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, China","institution_ids":["https://openalex.org/I16365422"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5090578084","display_name":"Xiaoshan Kai","orcid":"https://orcid.org/0000-0002-4384-1470"},"institutions":[{"id":"https://openalex.org/I16365422","display_name":"Hefei University of Technology","ror":"https://ror.org/02czkny70","country_code":"CN","type":"education","lineage":["https://openalex.org/I16365422"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Xiaoshan Kai","raw_affiliation_strings":["School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, China","institution_ids":["https://openalex.org/I16365422"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5135648557","display_name":"Shixin Zhu","orcid":null},"institutions":[{"id":"https://openalex.org/I16365422","display_name":"Hefei University of Technology","ror":"https://ror.org/02czkny70","country_code":"CN","type":"education","lineage":["https://openalex.org/I16365422"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shixin Zhu","raw_affiliation_strings":["School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, China","institution_ids":["https://openalex.org/I16365422"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5090578084"],"corresponding_institution_ids":["https://openalex.org/I16365422"],"apc_list":{"value":2530,"currency":"USD","value_usd":2530},"apc_paid":{"value":2530,"currency":"USD","value_usd":2530},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.55849115,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"114","issue":null,"first_page":"102844","last_page":"102844"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9053000211715698,"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":0.9053000211715698,"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":0.05739999935030937,"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/T10796","display_name":"Cooperative Communication and Network Coding","score":0.012199999764561653,"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/separable-space","display_name":"Separable space","score":0.5504000186920166},{"id":"https://openalex.org/keywords/vector-space","display_name":"Vector space","score":0.5098999738693237},{"id":"https://openalex.org/keywords/linear-code","display_name":"Linear code","score":0.5002999901771545},{"id":"https://openalex.org/keywords/code","display_name":"Code (set theory)","score":0.46399998664855957},{"id":"https://openalex.org/keywords/weight-distribution","display_name":"Weight distribution","score":0.4332999885082245},{"id":"https://openalex.org/keywords/minimum-distance","display_name":"Minimum distance","score":0.39010000228881836},{"id":"https://openalex.org/keywords/dual-code","display_name":"Dual code","score":0.387800008058548}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8087999820709229},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6363000273704529},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.5504000186920166},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5299000144004822},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.5098999738693237},{"id":"https://openalex.org/C2400350","wikidata":"https://www.wikidata.org/wiki/Q1752667","display_name":"Linear code","level":4,"score":0.5002999901771545},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.46399998664855957},{"id":"https://openalex.org/C25432639","wikidata":"https://www.wikidata.org/wiki/Q3711790","display_name":"Weight distribution","level":2,"score":0.4332999885082245},{"id":"https://openalex.org/C3018063660","wikidata":"https://www.wikidata.org/wiki/Q6865448","display_name":"Minimum distance","level":2,"score":0.39010000228881836},{"id":"https://openalex.org/C73666251","wikidata":"https://www.wikidata.org/wiki/Q5310189","display_name":"Dual code","level":5,"score":0.387800008058548},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.3598000109195709},{"id":"https://openalex.org/C169021753","wikidata":"https://www.wikidata.org/wiki/Q201972","display_name":"Reed\u2013Muller code","level":5,"score":0.35830000042915344},{"id":"https://openalex.org/C2096984","wikidata":"https://www.wikidata.org/wiki/Q245464","display_name":"Cyclic code","level":5,"score":0.31520000100135803},{"id":"https://openalex.org/C2778945305","wikidata":"https://www.wikidata.org/wiki/Q6865502","display_name":"Minimum weight","level":2,"score":0.27970001101493835},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.27250000834465027},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.26249998807907104},{"id":"https://openalex.org/C113709454","wikidata":"https://www.wikidata.org/wiki/Q602136","display_name":"Coding theory","level":2,"score":0.2565999925136566},{"id":"https://openalex.org/C49766605","wikidata":"https://www.wikidata.org/wiki/Q207643","display_name":"Linear map","level":2,"score":0.2515000104904175}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1016/j.ffa.2026.102844","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.ffa.2026.102844","pdf_url":null,"source":{"id":"https://openalex.org/S925225069","display_name":"Finite Fields and Their Applications","issn_l":"1071-5797","issn":["1071-5797","1090-2465"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Finite Fields and Their Applications","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1016/j.ffa.2026.102844","is_oa":true,"landing_page_url":"https://doi.org/10.1016/j.ffa.2026.102844","pdf_url":null,"source":{"id":"https://openalex.org/S925225069","display_name":"Finite Fields and Their Applications","issn_l":"1071-5797","issn":["1071-5797","1090-2465"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Finite Fields and Their Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320328119","display_name":"National University's Basic Research Foundation of China","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":46,"referenced_works":["https://openalex.org/W1519782116","https://openalex.org/W1530125006","https://openalex.org/W1553785754","https://openalex.org/W1975963044","https://openalex.org/W1976677460","https://openalex.org/W1984748596","https://openalex.org/W1993830711","https://openalex.org/W1994637991","https://openalex.org/W1997433240","https://openalex.org/W2012785774","https://openalex.org/W2016370111","https://openalex.org/W2018839402","https://openalex.org/W2023308455","https://openalex.org/W2026532517","https://openalex.org/W2041788275","https://openalex.org/W2061483889","https://openalex.org/W2063121374","https://openalex.org/W2067992506","https://openalex.org/W2079651774","https://openalex.org/W2131949172","https://openalex.org/W2167461394","https://openalex.org/W3022561029","https://openalex.org/W3038537849","https://openalex.org/W3085938014","https://openalex.org/W3094527023","https://openalex.org/W3119621083","https://openalex.org/W3175644973","https://openalex.org/W4214828742","https://openalex.org/W4289821394","https://openalex.org/W4292937362","https://openalex.org/W4308933491","https://openalex.org/W4311625003","https://openalex.org/W4322725230","https://openalex.org/W4365505990","https://openalex.org/W4375929133","https://openalex.org/W4377263704","https://openalex.org/W4379623567","https://openalex.org/W4388233531","https://openalex.org/W4388921248","https://openalex.org/W4389665371","https://openalex.org/W4392728280","https://openalex.org/W4398189935","https://openalex.org/W4401879605","https://openalex.org/W4401930244","https://openalex.org/W4404239159","https://openalex.org/W4413394839"],"related_works":[],"abstract_inverted_index":{"For":[0,82],"an":[1,110,143,181,193],"[":[2,144,182],"n":[3,27,90,102,145,161,183],",":[4,6,13,48,103,121,146,162,186,247],"k":[5,29,147,187],"d":[7,33],"]":[8,148,188],"q":[9,93,101,149,160,189,204,214],"linear":[10,85,112],"code":[11,86,113,130,151,173,241],"C":[12,18,24,38,43,49,60,66,87,114,120,131,137,152,174,242],"the":[14,63,127,171,209,239,249,265],"singleton":[15],"defect":[16],"of":[17,65,88,119,126,136,199,202,212,221,225,255],"is":[19,50,62,123],"defined":[20,109],"by":[21,207],"S":[22,36,41],"(":[23,37,42,116,133,176,244],")":[25,39,45,118,135,178,246],"=":[26,40,46],"\u2212":[28,32],"+":[30,184,205,215],"1":[31,47,185],".":[34,67,138,217],"If":[35],"\u22a5":[44,61],"called":[51],"a":[52,83,95,124,154,165],"near":[53],"maximum":[54],"distance":[55],"separable":[56],"(NMDS)":[57],"code,":[58],"where":[59],"dual":[64],"NMDS":[68,150,190,200,226,234,240,267],"codes":[69,201,211,227,235,258,268],"have":[70],"important":[71],"applications":[72],"in":[73],"finite":[74],"projective":[75],"geometries,":[76],"designs":[77],"and":[78,94,106,153,167,248,252],"secret":[79],"sharing":[80],"schemes.":[81],"given":[84],"length":[89,203,213],"over":[91],"F":[92,100,159],"nonzero":[96,155],"vector":[97,156],"u":[98,117,157,177,245],"\u2208":[99,158],"Sun,":[104],"Ding":[105],"Chen":[107],"[44]":[108],"extended":[111,129,172],"\u203e":[115,132,175,243],"which":[122],"generalisation":[125],"classical":[128],"-1":[134],"In":[139,231],"this":[140],"paper,":[141],"for":[142,170],"we":[163,195,262],"provide":[164],"sufficient":[166],"necessary":[168],"condition":[169],"to":[179],"be":[180],"code.":[191],"As":[192],"application,":[194],"construct":[196],"two":[197,223],"classes":[198,224],"3":[206],"extending":[208],"Roth-Lempel":[210,257],"2":[216],"The":[218],"weight":[219],"distribution":[220],"these":[222],"are":[228,236,259,269],"also":[229],"given.":[230],"addition,":[232],"more":[233],"obtained":[237],"from":[238],"covering":[250],"radius":[251],"deep":[253],"holes":[254],"some":[256],"determined.":[260],"Moreover,":[261],"prove":[263],"that":[264],"constructed":[266],"optimal":[270],"locally":[271],"recoverable":[272],"codes.":[273]},"counts_by_year":[],"updated_date":"2026-05-10T06:08:35.923825","created_date":"2026-05-09T00:00:00"}