{"id":"https://openalex.org/W4392904706","doi":"https://doi.org/10.1109/tit.2024.3370410","title":"Computing Efficiently a Parity-Check Matrix for \u2124<i> <sub>p</sub> <sup>s</sup> </i>-Additive Codes","display_name":"Computing Efficiently a Parity-Check Matrix for \u2124<i> <sub>p</sub> <sup>s</sup> </i>-Additive Codes","publication_year":2024,"publication_date":"2024-03-18","ids":{"openalex":"https://openalex.org/W4392904706","doi":"https://doi.org/10.1109/tit.2024.3370410"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2024.3370410","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tit.2024.3370410","pdf_url":"https://ieeexplore.ieee.org/ielx7/18/4667673/10471390.pdf","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":true,"oa_status":"bronze","oa_url":"https://ieeexplore.ieee.org/ielx7/18/4667673/10471390.pdf","any_repository_has_fulltext":true},"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":[{"id":"https://openalex.org/I123044942","display_name":"Universitat Aut\u00f2noma de Barcelona","ror":"https://ror.org/052g8jq94","country_code":"ES","type":"education","lineage":["https://openalex.org/I123044942"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Cristina Fern\u00e1ndez-C\u00f3rdoba","raw_affiliation_strings":["Department of Information and Communications Engineering, Universitat Aut&#x00F2;noma de Barcelona, Cerdanyola del Vall&#x00E8;s, Bellaterra, Spain","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, Bellaterra, Spain","institution_ids":["https://openalex.org/I123044942"]},{"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/A5083305829","display_name":"Adri\u00e1n Torres-Mart\u00edn","orcid":"https://orcid.org/0000-0002-2489-8930"},"institutions":[{"id":"https://openalex.org/I123044942","display_name":"Universitat Aut\u00f2noma de Barcelona","ror":"https://ror.org/052g8jq94","country_code":"ES","type":"education","lineage":["https://openalex.org/I123044942"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Adri\u00e1n Torres-Mart\u00edn","raw_affiliation_strings":["Department of Information and Communications Engineering, Universitat Aut&#x00F2;noma de Barcelona, Cerdanyola del Vall&#x00E8;s, Bellaterra, Spain","Department of Information and Communications Engineering, Universitat Aut&#x00F2;noma de Barcelona, Cerdanyola del Vall&#x00E8;s, Spain"],"raw_orcid":"https://orcid.org/0000-0002-2489-8930","affiliations":[{"raw_affiliation_string":"Department of Information and Communications Engineering, Universitat Aut&#x00F2;noma de Barcelona, Cerdanyola del Vall&#x00E8;s, Bellaterra, Spain","institution_ids":["https://openalex.org/I123044942"]},{"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":[{"id":"https://openalex.org/I123044942","display_name":"Universitat Aut\u00f2noma de Barcelona","ror":"https://ror.org/052g8jq94","country_code":"ES","type":"education","lineage":["https://openalex.org/I123044942"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Merc\u00e8 Villanueva","raw_affiliation_strings":["Department of Information and Communications Engineering, Universitat Aut&#x00F2;noma de Barcelona, Cerdanyola del Vall&#x00E8;s, Bellaterra, Spain","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, Bellaterra, Spain","institution_ids":["https://openalex.org/I123044942"]},{"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":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3055,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.60014617,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"71","issue":"5","first_page":"3447","last_page":"3456"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9997000098228455,"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":0.9997000098228455,"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/T10237","display_name":"Cryptography and Data Security","score":0.9976999759674072,"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/T11321","display_name":"Error Correcting Code Techniques","score":0.9962000250816345,"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/parity","display_name":"Parity (physics)","score":0.5008256435394287},{"id":"https://openalex.org/keywords/parity-check-matrix","display_name":"Parity-check matrix","score":0.4956146478652954},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4455069303512573},{"id":"https://openalex.org/keywords/matrix-algebra","display_name":"Matrix algebra","score":0.41534852981567383},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3999100625514984},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.38855981826782227},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.36916226148605347},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3517630398273468},{"id":"https://openalex.org/keywords/decoding-methods","display_name":"Decoding methods","score":0.31476449966430664},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.28327926993370056},{"id":"https://openalex.org/keywords/low-density-parity-check-code","display_name":"Low-density parity-check code","score":0.2608400583267212},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.12703421711921692},{"id":"https://openalex.org/keywords/atomic-physics","display_name":"Atomic physics","score":0.09064242243766785}],"concepts":[{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.5008256435394287},{"id":"https://openalex.org/C95925971","wikidata":"https://www.wikidata.org/wiki/Q3100414","display_name":"Parity-check matrix","level":4,"score":0.4956146478652954},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4455069303512573},{"id":"https://openalex.org/C2988995629","wikidata":"https://www.wikidata.org/wiki/Q2915729","display_name":"Matrix algebra","level":3,"score":0.41534852981567383},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3999100625514984},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.38855981826782227},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.36916226148605347},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3517630398273468},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.31476449966430664},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.28327926993370056},{"id":"https://openalex.org/C67692717","wikidata":"https://www.wikidata.org/wiki/Q187444","display_name":"Low-density parity-check code","level":3,"score":0.2608400583267212},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.12703421711921692},{"id":"https://openalex.org/C184779094","wikidata":"https://www.wikidata.org/wiki/Q26383","display_name":"Atomic physics","level":1,"score":0.09064242243766785},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tit.2024.3370410","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tit.2024.3370410","pdf_url":"https://ieeexplore.ieee.org/ielx7/18/4667673/10471390.pdf","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:ddd.uab.cat:293051","is_oa":true,"landing_page_url":"https://ddd.uab.cat/record/293051","pdf_url":"https://ddd.uab.cat/pub/artpub/2025/293051/IEEETraInfThe_a2025v71n5p3447iENG.pdf","source":{"id":"https://openalex.org/S4306400380","display_name":"Dip\u00f2sit Digital de Documents de la UAB (Universitat Aut\u00f2noma de Barcelona)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I123044942","host_organization_name":"Universitat Aut\u00f2noma de Barcelona","host_organization_lineage":["https://openalex.org/I123044942"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.1109/tit.2024.3370410","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tit.2024.3370410","pdf_url":"https://ieeexplore.ieee.org/ielx7/18/4667673/10471390.pdf","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"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1200330925","display_name":null,"funder_award_id":"PID2019-104664GB-I00","funder_id":"https://openalex.org/F4320321837","funder_display_name":"Ministerio de Econom\u00eda y Competitividad"},{"id":"https://openalex.org/G2262748287","display_name":null,"funder_award_id":"501100011033","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G2352338862","display_name":null,"funder_award_id":"2021 SGR 00643","funder_id":"https://openalex.org/F4320334830","funder_display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca"},{"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/G288263039","display_name":null,"funder_award_id":"10.13039/501100011033","funder_id":"https://openalex.org/F4320334830","funder_display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca"},{"id":"https://openalex.org/G300979063","display_name":null,"funder_award_id":"10.13039/501100011033","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G3069120617","display_name":null,"funder_award_id":"04106","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G3126258611","display_name":null,"funder_award_id":"PID2019-104664GB-I00","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G3461949579","display_name":null,"funder_award_id":"UIDB/04106/2020","funder_id":"https://openalex.org/F4320329951","funder_display_name":"Center for Research and Development in Mathematics and Applications"},{"id":"https://openalex.org/G3480869486","display_name":null,"funder_award_id":"13039","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G4158340598","display_name":null,"funder_award_id":"PID2022-137924NB-I00","funder_id":"https://openalex.org/F4320321837","funder_display_name":"Ministerio de Econom\u00eda y Competitividad"},{"id":"https://openalex.org/G4429097262","display_name":null,"funder_award_id":"AEI / 10.13039/501100011033","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G5080475149","display_name":null,"funder_award_id":"10.13039","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G5852091828","display_name":null,"funder_award_id":"13039/501100011033","funder_id":"https://openalex.org/F4320334830","funder_display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca"},{"id":"https://openalex.org/G6687440681","display_name":null,"funder_award_id":"ID2019-104664GB-I00","funder_id":"https://openalex.org/F4320321837","funder_display_name":"Ministerio de Econom\u00eda y Competitividad"},{"id":"https://openalex.org/G6808216599","display_name":null,"funder_award_id":"021 SGR 00643","funder_id":"https://openalex.org/F4320334830","funder_display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca"},{"id":"https://openalex.org/G7068753761","display_name":null,"funder_award_id":"RED2022-134306-T (AEI / 10.13039/501100011033)","funder_id":"https://openalex.org/F4320321837","funder_display_name":"Ministerio de Econom\u00eda y Competitividad"},{"id":"https://openalex.org/G7084143925","display_name":null,"funder_award_id":"AEI/10","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G7266728691","display_name":null,"funder_award_id":"13039/501100011033","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G7662128921","display_name":null,"funder_award_id":"UIDB/04106/2020","funder_id":"https://openalex.org/F4320310000","funder_display_name":"Universidade de Aveiro"},{"id":"https://openalex.org/G8665597936","display_name":null,"funder_award_id":"RED2022-134306-T (AEI/10.13039/ 501100011033)","funder_id":"https://openalex.org/F4320321837","funder_display_name":"Ministerio de Econom\u00eda y Competitividad"},{"id":"https://openalex.org/G8684520763","display_name":null,"funder_award_id":"PID2019","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G894250657","display_name":null,"funder_award_id":"ID2022-137924NB-I00","funder_id":"https://openalex.org/F4320321837","funder_display_name":"Ministerio de Econom\u00eda y Competitividad"}],"funders":[{"id":"https://openalex.org/F4320310000","display_name":"Universidade de Aveiro","ror":"https://ror.org/00nt41z93"},{"id":"https://openalex.org/F4320321837","display_name":"Ministerio de Econom\u00eda y Competitividad","ror":"https://ror.org/034900433"},{"id":"https://openalex.org/F4320326262","display_name":"Ministerio de Asuntos Econ\u00f3micos y Transformaci\u00f3n Digital, Gobierno de Espa\u00f1a","ror":"https://ror.org/03sv46s19"},{"id":"https://openalex.org/F4320329951","display_name":"Center for Research and Development in Mathematics and Applications","ror":null},{"id":"https://openalex.org/F4320334779","display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","ror":"https://ror.org/00snfqn58"},{"id":"https://openalex.org/F4320334830","display_name":"Ag\u00e8ncia de Gesti\u00f3 d'Ajuts Universitaris i de Recerca","ror":"https://ror.org/01n4pqe45"},{"id":"https://openalex.org/F4320335598","display_name":"Agencia Estatal de Investigaci\u00f3n","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4392904706.pdf","grobid_xml":"https://content.openalex.org/works/W4392904706.grobid-xml"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W194960238","https://openalex.org/W1516318957","https://openalex.org/W1619836270","https://openalex.org/W1964495070","https://openalex.org/W1973810401","https://openalex.org/W1981315250","https://openalex.org/W1987121126","https://openalex.org/W2010238614","https://openalex.org/W2038681062","https://openalex.org/W2062142782","https://openalex.org/W2085147719","https://openalex.org/W2109887866","https://openalex.org/W2121513675","https://openalex.org/W2151948311","https://openalex.org/W2171843169","https://openalex.org/W2395800580","https://openalex.org/W2739214207","https://openalex.org/W2953008024","https://openalex.org/W3201287454","https://openalex.org/W4205551177","https://openalex.org/W4214810519","https://openalex.org/W4225530725","https://openalex.org/W4388795554","https://openalex.org/W4401692196","https://openalex.org/W6682537286"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2472109360","https://openalex.org/W2161474341","https://openalex.org/W2145403642","https://openalex.org/W2118148379","https://openalex.org/W3080746295","https://openalex.org/W2125154953","https://openalex.org/W4293057822","https://openalex.org/W2140459655","https://openalex.org/W1835960588"],"abstract_inverted_index":{"The":[0,131],"Z":[1,15,43,48,57,76,133],"<italic":[2,9,16,22,27,77,134],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[3,10,17,23,28,45,50,59,78,135],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><sub>p<sup>s</sup></sub></i>":[4,79,136],"-additive":[5,80],"codes":[6,41,112],"of":[7,14,39,74,86,97,128,143],"length":[8],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">n</i>":[11],"are":[12],"subgroups":[13],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><sup>n</sup>":[18],"<sub>p<sup>s</sup></sub></i>":[19],",":[20,47,52],"with":[21,103],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">p</i>":[24],"prime":[25],"and":[26],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">s</i>":[29],"\u2265":[30],"1.":[31],"They":[32],"can":[33,157],"be":[34,158],"seen":[35],"as":[36],"a":[37,71,75,83,124,139,165],"generalization":[38],"linear":[40,111,162],"over":[42,56,113,164],"<sub":[44,49,58],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sub>":[46],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">4</sub>":[51],"or":[53],"more":[54,140],"general":[55,141],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2<sup><i>s</i></sup></sub>":[60],".":[61],"In":[62],"this":[63,119],"paper,":[64,150],"we":[65,121,151],"show":[66,123],"two":[67],"methods":[68],"for":[69,110],"computing":[70],"parity-check":[72],"matrix":[73,85],"code":[81,88,163],"from":[82],"generator":[84],"the":[87,95,104,129,149,154],"in":[89,101,108,116],"standard":[90],"form.":[91],"We":[92],"also":[93,122],"compare":[94],"performance":[96],"our":[98],"results":[99,156],"implemented":[100],"Magma":[102,109],"current":[105],"available":[106],"function":[107],"finite":[114,145,166],"rings":[115,132],"general.":[117],"Complementing":[118],"comparison,":[120],"time":[125],"complexity":[126],"analysis":[127],"algorithms.":[130],"belong":[137],"to":[138,160],"class":[142],"rings:":[144],"chain":[146,168],"rings.":[147],"Along":[148],"observe":[152],"that":[153],"same":[155],"applied":[159],"any":[161],"commutative":[167],"ring.":[169]},"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2024-03-19T00:00:00"}
