{"id":"https://openalex.org/W3038331066","doi":"https://doi.org/10.1142/s1793830920500858","title":"Linear codes over \ud835\udd3d4R and their MacWilliams identity","display_name":"Linear codes over \ud835\udd3d4R and their MacWilliams identity","publication_year":2020,"publication_date":"2020-07-01","ids":{"openalex":"https://openalex.org/W3038331066","doi":"https://doi.org/10.1142/s1793830920500858","mag":"3038331066"},"language":"en","primary_location":{"id":"doi:10.1142/s1793830920500858","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s1793830920500858","pdf_url":null,"source":{"id":"https://openalex.org/S178434951","display_name":"Discrete Mathematics Algorithms and Applications","issn_l":"1793-8309","issn":["1793-8309","1793-8317"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete Mathematics, Algorithms and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://dr.ntu.edu.sg/bitstream/10356/146385/2/Linear%20codes%20over%20F4R%20and%20their%20MacWilliams%20identity.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5027173344","display_name":"Nasreddine Benbelkacem","orcid":"https://orcid.org/0000-0002-0051-0276"},"institutions":[{"id":"https://openalex.org/I157169809","display_name":"University of Sciences and Technology Houari Boumediene","ror":"https://ror.org/02kb89c09","country_code":"DZ","type":"education","lineage":["https://openalex.org/I157169809"]},{"id":"https://openalex.org/I172675005","display_name":"Nanyang Technological University","ror":"https://ror.org/02e7b5302","country_code":"SG","type":"education","lineage":["https://openalex.org/I172675005"]},{"id":"https://openalex.org/I199440890","display_name":"American University of Sharjah","ror":"https://ror.org/001g2fj96","country_code":"AE","type":"education","lineage":["https://openalex.org/I199440890"]}],"countries":["AE","DZ","SG"],"is_corresponding":true,"raw_author_name":"Nasreddine Benbelkacem","raw_affiliation_strings":["Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El Alia, Bab Ezzouar, 16111 Algiers, Algeria","School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371","Department of Mathematics and Statistics, College of Arts and Sciences, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El Alia, Bab Ezzouar, 16111 Algiers, Algeria","institution_ids":["https://openalex.org/I157169809"]},{"raw_affiliation_string":"School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371","institution_ids":["https://openalex.org/I172675005"]},{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Arts and Sciences, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates","institution_ids":["https://openalex.org/I199440890"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026666126","display_name":"Martianus Frederic Ezerman","orcid":"https://orcid.org/0000-0002-5851-2717"},"institutions":[{"id":"https://openalex.org/I172675005","display_name":"Nanyang Technological University","ror":"https://ror.org/02e7b5302","country_code":"SG","type":"education","lineage":["https://openalex.org/I172675005"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Martianus Frederic Ezerman","raw_affiliation_strings":["School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371","institution_ids":["https://openalex.org/I172675005"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019610943","display_name":"Taher Abualrub","orcid":"https://orcid.org/0000-0002-3917-0862"},"institutions":[{"id":"https://openalex.org/I157169809","display_name":"University of Sciences and Technology Houari Boumediene","ror":"https://ror.org/02kb89c09","country_code":"DZ","type":"education","lineage":["https://openalex.org/I157169809"]},{"id":"https://openalex.org/I172675005","display_name":"Nanyang Technological University","ror":"https://ror.org/02e7b5302","country_code":"SG","type":"education","lineage":["https://openalex.org/I172675005"]},{"id":"https://openalex.org/I199440890","display_name":"American University of Sharjah","ror":"https://ror.org/001g2fj96","country_code":"AE","type":"education","lineage":["https://openalex.org/I199440890"]}],"countries":["AE","DZ","SG"],"is_corresponding":false,"raw_author_name":"Taher Abualrub","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Arts and Sciences, American University of Sharjah, P. O. Box 26666, Sharjah, United Arab Emirates","Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El Alia, Bab Ezzouar, 16111 Algiers, Algeria","School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371","Department of Mathematics and Statistics, College of Arts and Sciences, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates"],"raw_orcid":"https://orcid.org/0000-0002-3917-0862","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Arts and Sciences, American University of Sharjah, P. O. Box 26666, Sharjah, United Arab Emirates","institution_ids":["https://openalex.org/I199440890"]},{"raw_affiliation_string":"Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El Alia, Bab Ezzouar, 16111 Algiers, Algeria","institution_ids":["https://openalex.org/I157169809"]},{"raw_affiliation_string":"School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371","institution_ids":["https://openalex.org/I172675005"]},{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Arts and Sciences, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates","institution_ids":["https://openalex.org/I199440890"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5027173344"],"corresponding_institution_ids":["https://openalex.org/I157169809","https://openalex.org/I172675005","https://openalex.org/I199440890"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.07068124,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"12","issue":"06","first_page":"2050085","last_page":"2050085"},"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/T11797","display_name":"graph theory and CDMA systems","score":0.9937000274658203,"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":0.9836000204086304,"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/generator-matrix","display_name":"Generator matrix","score":0.7731586694717407},{"id":"https://openalex.org/keywords/identity","display_name":"Identity (music)","score":0.6788912415504456},{"id":"https://openalex.org/keywords/ring","display_name":"Ring (chemistry)","score":0.6391681432723999},{"id":"https://openalex.org/keywords/commutative-ring","display_name":"Commutative ring","score":0.6293275356292725},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5971929430961609},{"id":"https://openalex.org/keywords/generator","display_name":"Generator (circuit theory)","score":0.5690062046051025},{"id":"https://openalex.org/keywords/linear-code","display_name":"Linear code","score":0.5072655081748962},{"id":"https://openalex.org/keywords/dual","display_name":"Dual (grammatical number)","score":0.48562803864479065},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.4604083001613617},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.44994133710861206},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.41644617915153503},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4143662452697754},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.401240736246109},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33362340927124023},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.32916930317878723},{"id":"https://openalex.org/keywords/block-code","display_name":"Block code","score":0.21794584393501282},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.17341682314872742},{"id":"https://openalex.org/keywords/decoding-methods","display_name":"Decoding methods","score":0.08474743366241455},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.0680469274520874},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.06547939777374268},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.051775217056274414}],"concepts":[{"id":"https://openalex.org/C1842014","wikidata":"https://www.wikidata.org/wiki/Q1502046","display_name":"Generator matrix","level":3,"score":0.7731586694717407},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.6788912415504456},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.6391681432723999},{"id":"https://openalex.org/C161491579","wikidata":"https://www.wikidata.org/wiki/Q858656","display_name":"Commutative ring","level":3,"score":0.6293275356292725},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5971929430961609},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.5690062046051025},{"id":"https://openalex.org/C2400350","wikidata":"https://www.wikidata.org/wiki/Q1752667","display_name":"Linear code","level":4,"score":0.5072655081748962},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.48562803864479065},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.4604083001613617},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.44994133710861206},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.41644617915153503},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4143662452697754},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.401240736246109},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33362340927124023},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.32916930317878723},{"id":"https://openalex.org/C157125643","wikidata":"https://www.wikidata.org/wiki/Q884707","display_name":"Block code","level":3,"score":0.21794584393501282},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.17341682314872742},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.08474743366241455},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0680469274520874},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.06547939777374268},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.051775217056274414},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","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/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1142/s1793830920500858","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s1793830920500858","pdf_url":null,"source":{"id":"https://openalex.org/S178434951","display_name":"Discrete Mathematics Algorithms and Applications","issn_l":"1793-8309","issn":["1793-8309","1793-8317"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete Mathematics, Algorithms and Applications","raw_type":"journal-article"},{"id":"pmh:oai:dr.ntu.edu.sg:10356/146385","is_oa":true,"landing_page_url":"https://hdl.handle.net/10356/146385","pdf_url":"https://dr.ntu.edu.sg/bitstream/10356/146385/2/Linear%20codes%20over%20F4R%20and%20their%20MacWilliams%20identity.pdf","source":{"id":"https://openalex.org/S4306402609","display_name":"DR-NTU (Nanyang Technological University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I172675005","host_organization_name":"Nanyang Technological University","host_organization_lineage":["https://openalex.org/I172675005"],"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":"Journal Article"}],"best_oa_location":{"id":"pmh:oai:dr.ntu.edu.sg:10356/146385","is_oa":true,"landing_page_url":"https://hdl.handle.net/10356/146385","pdf_url":"https://dr.ntu.edu.sg/bitstream/10356/146385/2/Linear%20codes%20over%20F4R%20and%20their%20MacWilliams%20identity.pdf","source":{"id":"https://openalex.org/S4306402609","display_name":"DR-NTU (Nanyang Technological University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I172675005","host_organization_name":"Nanyang Technological University","host_organization_lineage":["https://openalex.org/I172675005"],"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":"Journal Article"},"sustainable_development_goals":[{"display_name":"Quality Education","id":"https://metadata.un.org/sdg/4","score":0.4300000071525574}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3038331066.pdf","grobid_xml":"https://content.openalex.org/works/W3038331066.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W597381772","https://openalex.org/W1588687207","https://openalex.org/W2010238614","https://openalex.org/W2071317810","https://openalex.org/W2234898970","https://openalex.org/W3128166402"],"related_works":["https://openalex.org/W4308298797","https://openalex.org/W3198007960","https://openalex.org/W4322759337","https://openalex.org/W4247927056","https://openalex.org/W2360590989","https://openalex.org/W4255963200","https://openalex.org/W2131338819","https://openalex.org/W4288278449","https://openalex.org/W2369074270","https://openalex.org/W2408222820"],"abstract_inverted_index":{"Let":[0],"[Formula:":[1,13,20,24,28,42,88],"see":[2,14,21,25,29,43,89],"text]":[3,15,22,26,44],"be":[4],"the":[5,16,37,59,63,68,76,81],"field":[6],"of":[7,39,62,67],"four":[8],"elements.":[9],"We":[10,56,74],"denote":[11],"by":[12,79],"commutative":[17],"ring,":[18],"with":[19,27],"elements,":[23],"text].":[30,90],"This":[31],"work":[32],"defines":[33],"linear":[34,85],"codes":[35,50,69,86],"over":[36,87],"ring":[38],"mixed":[40],"alphabets":[41],"as":[45,47],"well":[46],"their":[48,71],"dual":[49,72],"under":[51],"a":[52],"nondegenerate":[53],"inner":[54],"product.":[55],"then":[57],"derive":[58],"systematic":[60],"form":[61],"respective":[64],"generator":[65],"matrices":[66],"and":[70],"codes.":[73],"wrap":[75],"paper":[77],"up":[78],"proving":[80],"MacWilliams":[82],"identity":[83],"for":[84]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2020-07-10T00:00:00"}