{"id":"https://openalex.org/W2937048679","doi":"https://doi.org/10.1109/tit.2020.2971989","title":"Polynomial Invariant Theory and Shape Enumerator of Self-Dual Codes in the NRT-Metric","display_name":"Polynomial Invariant Theory and Shape Enumerator of Self-Dual Codes in the NRT-Metric","publication_year":2020,"publication_date":"2020-02-06","ids":{"openalex":"https://openalex.org/W2937048679","doi":"https://doi.org/10.1109/tit.2020.2971989","mag":"2937048679"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2020.2971989","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2020.2971989","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":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1904.04333","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5049658395","display_name":"Welington Santos","orcid":"https://orcid.org/0000-0002-9458-4363"},"institutions":[{"id":"https://openalex.org/I52418104","display_name":"Universidade Federal do Paran\u00e1","ror":"https://ror.org/05syd6y78","country_code":"BR","type":"education","lineage":["https://openalex.org/I52418104"]},{"id":"https://openalex.org/I75619284","display_name":"Universidade Positivo","ror":"https://ror.org/02d09a271","country_code":"BR","type":"education","lineage":["https://openalex.org/I75619284"]}],"countries":["BR"],"is_corresponding":true,"raw_author_name":"Welington Santos","raw_affiliation_strings":["Programa de P??s-Gradua????o em Matem??tica, Universidade Federal do Paran\u00e1, Curitiba, Brazil","Universidade Positivo, Curitiba, Brazil","Programa de P\u00f3s-Gradua\u00e7\u00e3o em Matem\u00e1tica, Universidade Federal do Paran\u00e1, Curitiba, Brazil"],"affiliations":[{"raw_affiliation_string":"Programa de P??s-Gradua????o em Matem??tica, Universidade Federal do Paran\u00e1, Curitiba, Brazil","institution_ids":["https://openalex.org/I52418104"]},{"raw_affiliation_string":"Universidade Positivo, Curitiba, Brazil","institution_ids":["https://openalex.org/I75619284"]},{"raw_affiliation_string":"Programa de P\u00f3s-Gradua\u00e7\u00e3o em Matem\u00e1tica, Universidade Federal do Paran\u00e1, Curitiba, Brazil","institution_ids":["https://openalex.org/I52418104"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5043562972","display_name":"Marcelo Muniz S. Alves","orcid":"https://orcid.org/0000-0002-6772-3575"},"institutions":[{"id":"https://openalex.org/I52418104","display_name":"Universidade Federal do Paran\u00e1","ror":"https://ror.org/05syd6y78","country_code":"BR","type":"education","lineage":["https://openalex.org/I52418104"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Marcelo Muniz Alves","raw_affiliation_strings":["Departamento de Matem??tica, Universidade Federal do Paran\u00e1, Curitiba, Brazil","Departamento de Matem\u00e1tica, Universidade Federal do Parana, Curitiba, Brazil"],"affiliations":[{"raw_affiliation_string":"Departamento de Matem??tica, Universidade Federal do Paran\u00e1, Curitiba, Brazil","institution_ids":["https://openalex.org/I52418104"]},{"raw_affiliation_string":"Departamento de Matem\u00e1tica, Universidade Federal do Parana, Curitiba, Brazil","institution_ids":["https://openalex.org/I52418104"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5049658395"],"corresponding_institution_ids":["https://openalex.org/I52418104","https://openalex.org/I75619284"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.00533966,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"66","issue":"7","first_page":"4061","last_page":"4074"},"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/T10849","display_name":"Finite Group Theory Research","score":0.9997000098228455,"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"}},{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9994999766349792,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.7097534537315369},{"id":"https://openalex.org/keywords/dual","display_name":"Dual (grammatical number)","score":0.6962851285934448},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6302031874656677},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.5702410936355591},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.49457356333732605},{"id":"https://openalex.org/keywords/invariant-theory","display_name":"Invariant theory","score":0.47944727540016174},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4776093661785126},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.44077202677726746},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.20513802766799927},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.2018488645553589},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.13773494958877563},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.0799303650856018}],"concepts":[{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.7097534537315369},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.6962851285934448},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6302031874656677},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.5702410936355591},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49457356333732605},{"id":"https://openalex.org/C63036529","wikidata":"https://www.wikidata.org/wiki/Q1855669","display_name":"Invariant theory","level":3,"score":0.47944727540016174},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4776093661785126},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.44077202677726746},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.20513802766799927},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2018488645553589},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.13773494958877563},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.0799303650856018},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","level":0,"score":0.0},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0},{"id":"https://openalex.org/C124952713","wikidata":"https://www.wikidata.org/wiki/Q8242","display_name":"Literature","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1109/tit.2020.2971989","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2020.2971989","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"},{"id":"pmh:oai:arXiv.org:1904.04333","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1904.04333","pdf_url":"https://arxiv.org/pdf/1904.04333","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:2937048679","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/1904.04333","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1904.04333","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1904.04333","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1904.04333","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1904.04333","pdf_url":"https://arxiv.org/pdf/1904.04333","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321091","display_name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","ror":"https://ror.org/00x0ma614"},{"id":"https://openalex.org/F4320322025","display_name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","ror":"https://ror.org/03swz6y49"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2937048679.pdf","grobid_xml":"https://content.openalex.org/works/W2937048679.grobid-xml"},"referenced_works_count":33,"referenced_works":["https://openalex.org/W27315300","https://openalex.org/W179392144","https://openalex.org/W1519801683","https://openalex.org/W1617868970","https://openalex.org/W1740055512","https://openalex.org/W1827451679","https://openalex.org/W1963661276","https://openalex.org/W1965656255","https://openalex.org/W1976677460","https://openalex.org/W2002698792","https://openalex.org/W2007416660","https://openalex.org/W2020465878","https://openalex.org/W2026824916","https://openalex.org/W2031242533","https://openalex.org/W2040371959","https://openalex.org/W2043474783","https://openalex.org/W2045023532","https://openalex.org/W2069047683","https://openalex.org/W2073483687","https://openalex.org/W2145201379","https://openalex.org/W2147588253","https://openalex.org/W2158309140","https://openalex.org/W2164748522","https://openalex.org/W2313163255","https://openalex.org/W2552716967","https://openalex.org/W2905778870","https://openalex.org/W2906680874","https://openalex.org/W2914079865","https://openalex.org/W2963056680","https://openalex.org/W2963361860","https://openalex.org/W6601077106","https://openalex.org/W6636427457","https://openalex.org/W6637725580"],"related_works":["https://openalex.org/W3005052728","https://openalex.org/W2060988076","https://openalex.org/W1595335682","https://openalex.org/W2963639234","https://openalex.org/W2952185030","https://openalex.org/W184065422","https://openalex.org/W2406904678","https://openalex.org/W2009346046","https://openalex.org/W2143423882","https://openalex.org/W1989867438","https://openalex.org/W2760825081","https://openalex.org/W2155647239","https://openalex.org/W2018360032","https://openalex.org/W2767540775","https://openalex.org/W2988740442","https://openalex.org/W1564330504","https://openalex.org/W2069670450","https://openalex.org/W2110518906","https://openalex.org/W2009125260","https://openalex.org/W2158309140"],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3,74,82],"consider":[4],"self-dual":[5,10,46,50,57,92,129,150],"NRT":[6,47,51,58,93,130,151],"codes,":[7],"that":[8,81],"is,":[9],"codes":[11,131,152],"in":[12,60,95],"the":[13,18,40,76,87,107,147],"metric":[14,20],"space":[15],"endowed":[16],"with":[17],"Niederreiter-Rosenbloom-Tsfasman":[19],"(NRT":[21],"metric)":[22],"and":[23,31,53,124],"their":[24],"shape":[25,41,88],"enumerators":[26],"as":[27],"defined":[28],"by":[29,71],"Barg":[30],"Park.":[32],"We":[33,105,138],"use":[34],"polynomial":[35],"invariant":[36,79],"theory":[37],"to":[38,85,146],"describe":[39,75,86],"enumerator":[42,89],"of":[43,78,90,110,128,143,149,153],"a":[44,91,111],"binary":[45],"code,":[48,52],"even":[49,56],"weak":[54],"doubly":[55],"code":[59,94],"M":[61,96,115],"<sub":[62,66,97,101,116,120,134],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[63,67,98,102,117,121,135],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n,2</sub>":[64],"(F":[65,100,119],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sub>":[68,103],").":[69,104],"Motivated":[70],"these":[72],"results,":[73],"number":[77],"polynomials":[80],"must":[83],"find":[84],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n,s</sub>":[99],"define":[106],"ordered":[108,144],"flip":[109,145],"matrix":[112],"A":[113],"\u2208":[114],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">k,ns</sub>":[118],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">q</sub>":[122,136],")":[123],"present":[125],"some":[126],"constructions":[127],"over":[132],"F":[133],".":[137],"further":[139],"give":[140],"an":[141],"application":[142],"classification":[148],"dimension":[154],"two.":[155]},"counts_by_year":[],"updated_date":"2026-03-10T16:38:18.471706","created_date":"2025-10-10T00:00:00"}