{"id":"https://openalex.org/W4408777522","doi":"https://doi.org/10.3390/e27040337","title":"On Algebraic Properties of Primitive Eisenstein Integers with Applications in Coding Theory","display_name":"On Algebraic Properties of Primitive Eisenstein Integers with Applications in Coding Theory","publication_year":2025,"publication_date":"2025-03-24","ids":{"openalex":"https://openalex.org/W4408777522","doi":"https://doi.org/10.3390/e27040337","pmid":"https://pubmed.ncbi.nlm.nih.gov/40282572"},"language":"en","primary_location":{"id":"doi:10.3390/e27040337","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e27040337","pdf_url":null,"source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/e27040337","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058635140","display_name":"Abdul Hadi","orcid":"https://orcid.org/0000-0001-6536-9894"},"institutions":[{"id":"https://openalex.org/I165230279","display_name":"Universitas Gadjah Mada","ror":"https://ror.org/03ke6d638","country_code":"ID","type":"education","lineage":["https://openalex.org/I165230279"]},{"id":"https://openalex.org/I92961808","display_name":"Riau University","ror":"https://ror.org/00nk7p507","country_code":"ID","type":"education","lineage":["https://openalex.org/I92961808"]}],"countries":["ID"],"is_corresponding":false,"raw_author_name":"Abdul Hadi","raw_affiliation_strings":["Department of Mathematics, Universitas Gadjah Mada, Sekip Utara Bulaksumur 21, Yogyakarta 55281, Indonesia","Department of Mathematics, Universitas Riau, Tampan, Pekanbaru 28293, Indonesia"],"raw_orcid":"https://orcid.org/0000-0001-6536-9894","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Universitas Gadjah Mada, Sekip Utara Bulaksumur 21, Yogyakarta 55281, Indonesia","institution_ids":["https://openalex.org/I165230279"]},{"raw_affiliation_string":"Department of Mathematics, Universitas Riau, Tampan, Pekanbaru 28293, Indonesia","institution_ids":["https://openalex.org/I92961808"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5009875226","display_name":"Uha Isnaini","orcid":"https://orcid.org/0000-0002-5313-813X"},"institutions":[{"id":"https://openalex.org/I165230279","display_name":"Universitas Gadjah Mada","ror":"https://ror.org/03ke6d638","country_code":"ID","type":"education","lineage":["https://openalex.org/I165230279"]}],"countries":["ID"],"is_corresponding":false,"raw_author_name":"Uha Isnaini","raw_affiliation_strings":["Department of Mathematics, Universitas Gadjah Mada, Sekip Utara Bulaksumur 21, Yogyakarta 55281, Indonesia"],"raw_orcid":"https://orcid.org/0000-0002-5313-813X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Universitas Gadjah Mada, Sekip Utara Bulaksumur 21, Yogyakarta 55281, Indonesia","institution_ids":["https://openalex.org/I165230279"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050182747","display_name":"Indah Emilia Wijayanti","orcid":"https://orcid.org/0000-0003-0390-8682"},"institutions":[{"id":"https://openalex.org/I165230279","display_name":"Universitas Gadjah Mada","ror":"https://ror.org/03ke6d638","country_code":"ID","type":"education","lineage":["https://openalex.org/I165230279"]}],"countries":["ID"],"is_corresponding":false,"raw_author_name":"Indah Emilia Wijayanti","raw_affiliation_strings":["Department of Mathematics, Universitas Gadjah Mada, Sekip Utara Bulaksumur 21, Yogyakarta 55281, Indonesia"],"raw_orcid":"https://orcid.org/0000-0003-0390-8682","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Universitas Gadjah Mada, Sekip Utara Bulaksumur 21, Yogyakarta 55281, Indonesia","institution_ids":["https://openalex.org/I165230279"]}]},{"author_position":"last","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":true,"raw_author_name":"Martianus Frederic Ezerman","raw_affiliation_strings":["School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore"],"raw_orcid":"https://orcid.org/0000-0002-5851-2717","affiliations":[{"raw_affiliation_string":"School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore","institution_ids":["https://openalex.org/I172675005"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5026666126"],"corresponding_institution_ids":["https://openalex.org/I172675005"],"apc_list":{"value":2600,"currency":"CHF","value_usd":3137},"apc_paid":{"value":2600,"currency":"CHF","value_usd":3137},"fwci":2.2618,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.88637897,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":96},"biblio":{"volume":"27","issue":"4","first_page":"337","last_page":"337"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9818000197410583,"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/T12162","display_name":"Cellular Automata and Applications","score":0.9710999727249146,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/eisenstein-integer","display_name":"Eisenstein integer","score":0.8835244178771973},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8123167753219604},{"id":"https://openalex.org/keywords/ring-of-integers","display_name":"Ring of integers","score":0.581002950668335},{"id":"https://openalex.org/keywords/modulo","display_name":"Modulo","score":0.5714595317840576},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5316404104232788},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.5089058876037598},{"id":"https://openalex.org/keywords/eisenstein-series","display_name":"Eisenstein series","score":0.5031206011772156},{"id":"https://openalex.org/keywords/coprime-integers","display_name":"Coprime integers","score":0.43868163228034973},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.43774229288101196},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.42190998792648315},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.41065335273742676},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.34878677129745483},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3233410716056824},{"id":"https://openalex.org/keywords/algebraic-number-field","display_name":"Algebraic number field","score":0.30259472131729126},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2652168571949005},{"id":"https://openalex.org/keywords/modular-form","display_name":"Modular form","score":0.13053229451179504},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09431236982345581}],"concepts":[{"id":"https://openalex.org/C38853081","wikidata":"https://www.wikidata.org/wiki/Q262370","display_name":"Eisenstein integer","level":3,"score":0.8835244178771973},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8123167753219604},{"id":"https://openalex.org/C164810661","wikidata":"https://www.wikidata.org/wiki/Q1358313","display_name":"Ring of integers","level":3,"score":0.581002950668335},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.5714595317840576},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5316404104232788},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.5089058876037598},{"id":"https://openalex.org/C133835828","wikidata":"https://www.wikidata.org/wiki/Q1313257","display_name":"Eisenstein series","level":3,"score":0.5031206011772156},{"id":"https://openalex.org/C23230895","wikidata":"https://www.wikidata.org/wiki/Q104752","display_name":"Coprime integers","level":2,"score":0.43868163228034973},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.43774229288101196},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42190998792648315},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.41065335273742676},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.34878677129745483},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3233410716056824},{"id":"https://openalex.org/C12657307","wikidata":"https://www.wikidata.org/wiki/Q616608","display_name":"Algebraic number field","level":2,"score":0.30259472131729126},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2652168571949005},{"id":"https://openalex.org/C75764964","wikidata":"https://www.wikidata.org/wiki/Q870797","display_name":"Modular form","level":2,"score":0.13053229451179504},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09431236982345581},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e27040337","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e27040337","pdf_url":null,"source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},{"id":"pmid:40282572","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/40282572","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":"Journal Article"},{"id":"pmh:oai:dr.ntu.edu.sg:10356/202226","is_oa":true,"landing_page_url":"https://hdl.handle.net/10356/202226","pdf_url":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":null,"raw_type":"Journal Article"},{"id":"pmh:oai:pubmedcentral.nih.gov:12025671","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/12025671","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Entropy (Basel)","raw_type":"Text"},{"id":"pmh:oai:doaj.org/article:9c9d4b83514845fb8a3f197e0883cffa","is_oa":true,"landing_page_url":"https://doaj.org/article/9c9d4b83514845fb8a3f197e0883cffa","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 27, Iss 4, p 337 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/e27040337","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e27040337","pdf_url":null,"source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W1550142585","https://openalex.org/W1966244415","https://openalex.org/W2004973703","https://openalex.org/W2021891144","https://openalex.org/W2029466501","https://openalex.org/W2052753485","https://openalex.org/W2097272744","https://openalex.org/W2150326445","https://openalex.org/W2153739324","https://openalex.org/W2159245467","https://openalex.org/W2513460785","https://openalex.org/W2556728283","https://openalex.org/W2562805420","https://openalex.org/W2569741146","https://openalex.org/W2765255838","https://openalex.org/W2887233996","https://openalex.org/W2888009369","https://openalex.org/W2888064251","https://openalex.org/W2891387165","https://openalex.org/W2938979467","https://openalex.org/W2990872506","https://openalex.org/W3164354906","https://openalex.org/W3196145622","https://openalex.org/W3198071097","https://openalex.org/W4214772016","https://openalex.org/W4233185505","https://openalex.org/W4313348277","https://openalex.org/W4391449877","https://openalex.org/W6637807268","https://openalex.org/W6829849190"],"related_works":["https://openalex.org/W2753617496","https://openalex.org/W3207146599","https://openalex.org/W3185238499","https://openalex.org/W4388566643","https://openalex.org/W2051123510","https://openalex.org/W3014764882","https://openalex.org/W2326425459","https://openalex.org/W2095243145","https://openalex.org/W2021409800","https://openalex.org/W3000319914"],"abstract_inverted_index":{"An":[0,22],"even":[1],"Eisenstein":[2,9,17,23,52,72,89,130],"integer":[3,18,24,29],"is":[4,19,26,34],"a":[5,55,84,93,115],"multiple":[6,30],"of":[7,11,31,67,80,88,105,113,128],"an":[8,16,28],"prime":[10,116],"the":[12,78,102,106,110,125,129,133],"least":[13],"norm.":[14],"Otherwise,":[15],"called":[20],"odd.":[21],"that":[25],"not":[27],"another":[32],"one":[33],"said":[35],"to":[36,44,91,119],"be":[37,42],"primitive.":[38],"Such":[39],"integers":[40,53,90],"can":[41],"used":[43],"construct":[45],"signal":[46],"constellations":[47],"and":[48,70],"complex-valued":[49],"codes":[50],"over":[51],"via":[54],"carefully":[56],"designed":[57],"modulo":[58,121],"function.":[59],"In":[60],"this":[61],"work,":[62],"we":[63],"establish":[64],"algebraic":[65],"properties":[66],"even,":[68],"odd,":[69],"primitive":[71],"integers.":[73],"We":[74,96],"investigate":[75],"conditions":[76],"for":[77],"set":[79,98],"all":[81],"units":[82],"in":[83,132],"given":[85],"quotient":[86],"ring":[87],"form":[92],"cyclic":[94],"group.":[95],"perform":[97],"partitioning":[99,112],"based":[100,123],"on":[101,124],"multiplicative":[103,126],"group":[104,127],"set.":[107],"This":[108],"generalizes":[109],"known":[111],"size":[114],"number":[117],"congruent":[118],"1":[120],"3":[122],"field":[131],"literature.":[134]},"counts_by_year":[{"year":2025,"cited_by_count":2}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}