{"id":"https://openalex.org/W4404045320","doi":"https://doi.org/10.3390/sym16111455","title":"Frobenius Local Rings of Order p4m","display_name":"Frobenius Local Rings of Order p4m","publication_year":2024,"publication_date":"2024-11-02","ids":{"openalex":"https://openalex.org/W4404045320","doi":"https://doi.org/10.3390/sym16111455"},"language":"en","primary_location":{"id":"doi:10.3390/sym16111455","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/sym16111455","pdf_url":null,"source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"http://dx.doi.org/10.3390/sym16111455","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5022321883","display_name":"Alhanouf Ali Alhomaidhi","orcid":"https://orcid.org/0009-0002-2520-2699"},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Alhanouf Ali Alhomaidhi","raw_affiliation_strings":["Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"],"raw_orcid":"https://orcid.org/0009-0002-2520-2699","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079457395","display_name":"Sami Alabiad","orcid":"https://orcid.org/0000-0002-6824-6985"},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Sami Alabiad","raw_affiliation_strings":["Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-6824-6985","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5048511358","display_name":"Nawal A. Alsarori","orcid":"https://orcid.org/0000-0002-9691-4979"},"institutions":[{"id":"https://openalex.org/I165259314","display_name":"Dr. Babasaheb Ambedkar Marathwada University","ror":"https://ror.org/033pfj584","country_code":"IN","type":"education","lineage":["https://openalex.org/I165259314"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Nawal A. Alsarori","raw_affiliation_strings":["Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India"],"raw_orcid":"https://orcid.org/0000-0002-9691-4979","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India","institution_ids":["https://openalex.org/I165259314"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5079457395"],"corresponding_institution_ids":["https://openalex.org/I28022161"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.16764842,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"16","issue":"11","first_page":"1455","last_page":"1455"},"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":0.9990000128746033,"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":0.9977999925613403,"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/mathematics","display_name":"Mathematics","score":0.8080241680145264},{"id":"https://openalex.org/keywords/local-ring","display_name":"Local ring","score":0.7255452871322632},{"id":"https://openalex.org/keywords/isomorphism","display_name":"Isomorphism (crystallography)","score":0.5285964012145996},{"id":"https://openalex.org/keywords/coprime-integers","display_name":"Coprime integers","score":0.5172077417373657},{"id":"https://openalex.org/keywords/commutative-ring","display_name":"Commutative ring","score":0.5144324898719788},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.4962514042854309},{"id":"https://openalex.org/keywords/residue-field","display_name":"Residue field","score":0.48874568939208984},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.48385632038116455},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.4771277904510498},{"id":"https://openalex.org/keywords/ring","display_name":"Ring (chemistry)","score":0.4494733512401581},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.42039114236831665},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4015585482120514},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.35831162333488464},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.33058810234069824}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8080241680145264},{"id":"https://openalex.org/C142109727","wikidata":"https://www.wikidata.org/wiki/Q1142704","display_name":"Local ring","level":3,"score":0.7255452871322632},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.5285964012145996},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.5172077417373657},{"id":"https://openalex.org/C161491579","wikidata":"https://www.wikidata.org/wiki/Q858656","display_name":"Commutative ring","level":3,"score":0.5144324898719788},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.4962514042854309},{"id":"https://openalex.org/C2781025942","wikidata":"https://www.wikidata.org/wiki/Q7315530","display_name":"Residue field","level":3,"score":0.48874568939208984},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.48385632038116455},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.4771277904510498},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.4494733512401581},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.42039114236831665},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4015585482120514},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.35831162333488464},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.33058810234069824},{"id":"https://openalex.org/C115624301","wikidata":"https://www.wikidata.org/wiki/Q895901","display_name":"Crystal structure","level":2,"score":0.0},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym16111455","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/sym16111455","pdf_url":null,"source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:50851927dd7d44048aa756ec7736f19a","is_oa":false,"landing_page_url":"https://doaj.org/article/50851927dd7d44048aa756ec7736f19a","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 16, Iss 11, p 1455 (2024)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/16/11/1455/","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16111455","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym16111455","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/sym16111455","pdf_url":null,"source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":15,"referenced_works":["https://openalex.org/W1548944036","https://openalex.org/W1973174922","https://openalex.org/W1976207490","https://openalex.org/W2046471908","https://openalex.org/W2068263566","https://openalex.org/W2070084344","https://openalex.org/W2083873712","https://openalex.org/W2085147719","https://openalex.org/W2519564756","https://openalex.org/W2601345000","https://openalex.org/W4239191450","https://openalex.org/W4394571507","https://openalex.org/W4402311227","https://openalex.org/W4402988657","https://openalex.org/W6735582636"],"related_works":["https://openalex.org/W2798939953","https://openalex.org/W2017162564","https://openalex.org/W3100652480","https://openalex.org/W2262537770","https://openalex.org/W2949907459","https://openalex.org/W4300973188","https://openalex.org/W2068614797","https://openalex.org/W4293771735","https://openalex.org/W2036976233","https://openalex.org/W4298929891"],"abstract_inverted_index":{"Suppose":[0],"R":[1,13],"is":[2,10,26],"a":[3,27,54,67],"finite":[4,119,122],"commutative":[5],"local":[6,41,59],"ring,":[7],"then":[8],"it":[9,52],"known":[11],"that":[12,98],"has":[14],"four":[15],"positive":[16],"integers":[17],"p,n,m,k":[18],"called":[19],"the":[20,33,71,96,104,108],"invariants":[21,81],"of":[22,40,48,57,61,70,73],"R,":[23],"where":[24],"p":[25],"prime":[28],"number.":[29],"This":[30,92],"paper":[31],"investigates":[32],"structure":[34],"and":[35,47,107],"classification":[36],"up":[37],"to":[38,79,121],"isomorphism":[39],"rings":[42,60,76,85],"with":[43,77],"residue":[44],"field":[45],"Fpm":[46],"length":[49],"4.":[50],"Specifically,":[51],"gives":[53],"comprehensive":[55],"characterization":[56],"Frobenius":[58,84,123],"order":[62],"p4m.":[63],"Furthermore,":[64],"we":[65],"provide":[66],"detailed":[68],"enumeration":[69],"classes":[72],"all":[74],"such":[75],"respect":[78],"their":[80],"p,n,m,k.":[82],"Finite":[83],"are":[86],"particularly":[87],"advantageous":[88],"for":[89,111],"coding":[90],"theory.":[91],"suitability":[93],"arises":[94],"from":[95,118],"fact":[97],"two":[99],"classical":[100],"theorems":[101],"by":[102],"MacWilliams,":[103],"Extension":[105],"Theorem":[106],"MacWilliams":[109],"relations":[110],"symmetrized":[112],"weight":[113],"enumerators,":[114],"can":[115],"be":[116],"generalized":[117],"fields":[120],"rings.":[124]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2024-11-05T00:00:00"}
