{"id":"https://openalex.org/W4285215921","doi":"https://doi.org/10.1109/tit.2022.3172884","title":"Nonlinearity and Kernel of Z-Linear Simplex and MacDonald Codes<sub> <sup> <i/> </sup> </sub>","display_name":"Nonlinearity and Kernel of Z-Linear Simplex and MacDonald Codes<sub> <sup> <i/> </sup> </sub>","publication_year":2022,"publication_date":"2022-05-04","ids":{"openalex":"https://openalex.org/W4285215921","doi":"https://doi.org/10.1109/tit.2022.3172884"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2022.3172884","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3172884","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/A5065292754","display_name":"Cristina Fern\u00e1ndez-C\u00f3rdoba","orcid":"https://orcid.org/0000-0001-5880-144X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cristina Fernandez-Cordoba","raw_affiliation_strings":["Department of Information and Communications Engineering, Universitat Aut&#x00F2;noma de Barcelona, Cerdanyola del Vall&#x00E8;s, Spain"],"raw_orcid":"https://orcid.org/0000-0001-5880-144X","affiliations":[{"raw_affiliation_string":"Department of Information and Communications Engineering, Universitat Aut&#x00F2;noma de Barcelona, Cerdanyola del Vall&#x00E8;s, Spain","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055663506","display_name":"Carlos Vela","orcid":"https://orcid.org/0000-0003-3362-8817"},"institutions":[{"id":"https://openalex.org/I60858718","display_name":"University of Aveiro","ror":"https://ror.org/00nt41z93","country_code":"PT","type":"education","lineage":["https://openalex.org/I60858718"]}],"countries":["PT"],"is_corresponding":false,"raw_author_name":"Carlos Vela","raw_affiliation_strings":["Department of Mathematics, University of Aveiro, Aveiro, Portugal"],"raw_orcid":"https://orcid.org/0000-0003-3362-8817","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Aveiro, Aveiro, Portugal","institution_ids":["https://openalex.org/I60858718"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5080668151","display_name":"Merc\u00e8 Villanueva","orcid":"https://orcid.org/0000-0001-6179-0833"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Merce Villanueva","raw_affiliation_strings":["Department of Information and Communications Engineering, Universitat Aut&#x00F2;noma de Barcelona, Cerdanyola del Vall&#x00E8;s, Spain"],"raw_orcid":"https://orcid.org/0000-0001-6179-0833","affiliations":[{"raw_affiliation_string":"Department of Information and Communications Engineering, Universitat Aut&#x00F2;noma de Barcelona, Cerdanyola del Vall&#x00E8;s, Spain","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.4162,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.6779165,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"68","issue":"11","first_page":"7174","last_page":"7183"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9994000196456909,"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":0.9994000196456909,"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/T10187","display_name":"Radio Frequency Integrated Circuit Design","score":0.957099974155426,"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/T11321","display_name":"Error Correcting Code Techniques","score":0.9422000050544739,"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/notation","display_name":"Notation","score":0.5704379081726074},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4511793851852417},{"id":"https://openalex.org/keywords/code","display_name":"Code (set theory)","score":0.4109358787536621},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3626056909561157},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36186784505844116},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3266831040382385},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.22201958298683167},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.17527654767036438},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.16416165232658386},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.0998275876045227}],"concepts":[{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.5704379081726074},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4511793851852417},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.4109358787536621},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3626056909561157},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36186784505844116},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3266831040382385},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.22201958298683167},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.17527654767036438},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.16416165232658386},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0998275876045227},{"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.2022.3172884","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3172884","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/G2590986966","display_name":"Center for Research and Development in Mathematics and Applications","funder_award_id":"UIDB/04106/2020","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G4017095837","display_name":null,"funder_award_id":"AEI/10.13039/501100011033","funder_id":"https://openalex.org/F4320322930","funder_display_name":"Ministerio de Ciencia e Innovaci\u00f3n"},{"id":"https://openalex.org/G5434187397","display_name":null,"funder_award_id":"PID2019-104664GB-I00","funder_id":"https://openalex.org/F4320322930","funder_display_name":"Ministerio de Ciencia e Innovaci\u00f3n"}],"funders":[{"id":"https://openalex.org/F4320322930","display_name":"Ministerio de Ciencia e Innovaci\u00f3n","ror":"https://ror.org/034900433"},{"id":"https://openalex.org/F4320334779","display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","ror":"https://ror.org/00snfqn58"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W1503013413","https://openalex.org/W1516318957","https://openalex.org/W1579261672","https://openalex.org/W1606480398","https://openalex.org/W1683886983","https://openalex.org/W1964495070","https://openalex.org/W1976673628","https://openalex.org/W1987121126","https://openalex.org/W2015797376","https://openalex.org/W2038681062","https://openalex.org/W2047094327","https://openalex.org/W2062142782","https://openalex.org/W2064737342","https://openalex.org/W2071383517","https://openalex.org/W2103971511","https://openalex.org/W2109887866","https://openalex.org/W2146744191","https://openalex.org/W2395800580","https://openalex.org/W2892003982","https://openalex.org/W2937251225","https://openalex.org/W2962972548","https://openalex.org/W2963946634","https://openalex.org/W2990636281","https://openalex.org/W2991239327","https://openalex.org/W2993272705","https://openalex.org/W3048948753","https://openalex.org/W3196378311","https://openalex.org/W4211147273","https://openalex.org/W4300529123","https://openalex.org/W6634985997"],"related_works":["https://openalex.org/W1978042415","https://openalex.org/W3086542228","https://openalex.org/W2017331178","https://openalex.org/W2976797620","https://openalex.org/W347180939","https://openalex.org/W3046926816","https://openalex.org/W3083539681","https://openalex.org/W2045490848","https://openalex.org/W2530212042","https://openalex.org/W2332694326"],"abstract_inverted_index":{"<inline-formula":[0,12,31,39,48,72,83,95,103,116,134,159,192,205],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[1,13,32,40,49,73,84,96,104,117,135,160,193,206],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">":[2,14,33,41,50,74,85,97,105,118,136,161,194,207],"<tex-math":[3,15,34,42,51,75,86,98,106,119,137,162,195,208],"notation=\"LaTeX\">$\\mathbb":[4,16,35,43,52,76,87,120,138,163,196],"{Z}_{2^{s}}$":[5,53,77,88,121,139,164,197],"</tex-math></inline-formula>":[6,18,37,45,54,78,89,101,109,122,140,165,198,210],"-additive":[7,79,90,141,199],"codes":[8,29,92,129,175,201],"are":[9,112,130,202,215],"subgroups":[10],"of":[11,27,70,93,123,156,173,178,188],"{Z}^{n}_{2^{s}}$":[17],",":[19,110,211],"and":[20,38,102,170,212],"can":[21],"be":[22],"seen":[23],"as":[24,176],"a":[25,58,71,113,153,171],"generalization":[26,114],"linear":[28,154],"over":[30,115],"{Z}_{2}$":[36],"{Z}_{4}$":[44],".":[46],"A":[47],"-linear":[55,166],"code":[56,60],"is":[57,65],"binary":[59,125],"(not":[61],"necessarily":[62],"linear)":[63],"which":[64,111],"the":[66,124,133,157,186],"Gray":[67],"map":[68],"image":[69],"code.":[80],"We":[81],"consider":[82],"simplex":[91,126],"type":[94],"notation=\"LaTeX\">$\\alpha":[99],"$":[100,108],"notation=\"LaTeX\">$\\beta":[107],"codes.":[127,143,190],"These":[128],"related":[131],"to":[132,151],"Hadamard":[142],"In":[144,181],"this":[145,149,179,183],"paper,":[146],"we":[147],"use":[148],"relationship":[150],"find":[152],"subcode":[155],"corresponding":[158],"codes,":[167],"called":[168],"kernel,":[169],"representation":[172],"these":[174,189],"cosets":[177],"kernel.":[180],"particular,":[182],"also":[184],"gives":[185],"linearity":[187],"Similarly,":[191],"MacDonald":[200],"defined":[203],"for":[204],"notation=\"LaTeX\">$s&gt;2$":[209],"equivalent":[213],"results":[214],"obtained.":[216]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
