{"id":"https://openalex.org/W2339231084","doi":"https://doi.org/10.1080/00207160.2015.1069819","title":"On left quaternion codes","display_name":"On left quaternion codes","publication_year":2015,"publication_date":"2015-07-09","ids":{"openalex":"https://openalex.org/W2339231084","doi":"https://doi.org/10.1080/00207160.2015.1069819","mag":"2339231084"},"language":"en","primary_location":{"id":"doi:10.1080/00207160.2015.1069819","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2015.1069819","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","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/A5100400900","display_name":"Yuan Cao","orcid":"https://orcid.org/0000-0002-3089-0046"},"institutions":[{"id":"https://openalex.org/I119203015","display_name":"Shandong University of Technology","ror":"https://ror.org/02mr3ar13","country_code":"CN","type":"education","lineage":["https://openalex.org/I119203015"]},{"id":"https://openalex.org/I16609230","display_name":"Hunan University","ror":"https://ror.org/05htk5m33","country_code":"CN","type":"education","lineage":["https://openalex.org/I16609230"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuan Cao","raw_affiliation_strings":["College of Mathematics and Econometrics, Hunan University, Changsha410082, China","School of Sciences, Shandong University of Technology, Zibo, Shandong255091, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and Econometrics, Hunan University, Changsha410082, China","institution_ids":["https://openalex.org/I16609230"]},{"raw_affiliation_string":"School of Sciences, Shandong University of Technology, Zibo, Shandong255091, China","institution_ids":["https://openalex.org/I119203015"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100400900","display_name":"Yuan Cao","orcid":"https://orcid.org/0000-0002-3089-0046"},"institutions":[{"id":"https://openalex.org/I119203015","display_name":"Shandong University of Technology","ror":"https://ror.org/02mr3ar13","country_code":"CN","type":"education","lineage":["https://openalex.org/I119203015"]},{"id":"https://openalex.org/I16609230","display_name":"Hunan University","ror":"https://ror.org/05htk5m33","country_code":"CN","type":"education","lineage":["https://openalex.org/I16609230"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yuan Cao","raw_affiliation_strings":["College of Mathematics and Econometrics, Hunan University, Changsha410082, China","School of Sciences, Shandong University of Technology, Zibo, Shandong255091, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and Econometrics, Hunan University, Changsha410082, China","institution_ids":["https://openalex.org/I16609230"]},{"raw_affiliation_string":"School of Sciences, Shandong University of Technology, Zibo, Shandong255091, China","institution_ids":["https://openalex.org/I119203015"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056649812","display_name":"Yonglin Cao","orcid":"https://orcid.org/0000-0002-3682-6483"},"institutions":[{"id":"https://openalex.org/I119203015","display_name":"Shandong University of Technology","ror":"https://ror.org/02mr3ar13","country_code":"CN","type":"education","lineage":["https://openalex.org/I119203015"]},{"id":"https://openalex.org/I16609230","display_name":"Hunan University","ror":"https://ror.org/05htk5m33","country_code":"CN","type":"education","lineage":["https://openalex.org/I16609230"]},{"id":"https://openalex.org/I205237279","display_name":"Nankai University","ror":"https://ror.org/01y1kjr75","country_code":"CN","type":"education","lineage":["https://openalex.org/I205237279"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yonglin Cao","raw_affiliation_strings":["College of Mathematics and Econometrics, Hunan University, Changsha410082, China","School of Sciences, Shandong University of Technology, Zibo, Shandong255091, China","Chern Institute of Mathematics and LPMC, Nankai University, Tianjin300071, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and Econometrics, Hunan University, Changsha410082, China","institution_ids":["https://openalex.org/I16609230"]},{"raw_affiliation_string":"School of Sciences, Shandong University of Technology, Zibo, Shandong255091, China","institution_ids":["https://openalex.org/I119203015"]},{"raw_affiliation_string":"Chern Institute of Mathematics and LPMC, Nankai University, Tianjin300071, China","institution_ids":["https://openalex.org/I205237279"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056649812","display_name":"Yonglin Cao","orcid":"https://orcid.org/0000-0002-3682-6483"},"institutions":[{"id":"https://openalex.org/I119203015","display_name":"Shandong University of Technology","ror":"https://ror.org/02mr3ar13","country_code":"CN","type":"education","lineage":["https://openalex.org/I119203015"]},{"id":"https://openalex.org/I16609230","display_name":"Hunan University","ror":"https://ror.org/05htk5m33","country_code":"CN","type":"education","lineage":["https://openalex.org/I16609230"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yonglin Cao","raw_affiliation_strings":["College of Mathematics and Econometrics, Hunan University, Changsha410082, China","School of Sciences, Shandong University of Technology, Zibo, Shandong255091, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and Econometrics, Hunan University, Changsha410082, China","institution_ids":["https://openalex.org/I16609230"]},{"raw_affiliation_string":"School of Sciences, Shandong University of Technology, Zibo, Shandong255091, China","institution_ids":["https://openalex.org/I119203015"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5063946169","display_name":"Fang\u2010Wei Fu","orcid":"https://orcid.org/0000-0002-9696-8977"},"institutions":[{"id":"https://openalex.org/I205237279","display_name":"Nankai University","ror":"https://ror.org/01y1kjr75","country_code":"CN","type":"education","lineage":["https://openalex.org/I205237279"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fang-Wei Fu","raw_affiliation_strings":["Chern Institute of Mathematics and LPMC, Nankai University, Tianjin300071, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Chern Institute of Mathematics and LPMC, Nankai University, Tianjin300071, China","institution_ids":["https://openalex.org/I205237279"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5056649812","https://openalex.org/A5100400900"],"corresponding_institution_ids":["https://openalex.org/I119203015","https://openalex.org/I16609230","https://openalex.org/I205237279"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.12609762,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"93","issue":"10","first_page":"1629","last_page":"1649"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9979000091552734,"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/T10796","display_name":"Cooperative Communication and Network Coding","score":0.9947999715805054,"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/mathematics","display_name":"Mathematics","score":0.756916880607605},{"id":"https://openalex.org/keywords/cyclic-code","display_name":"Cyclic code","score":0.6160991191864014},{"id":"https://openalex.org/keywords/quaternion","display_name":"Quaternion","score":0.5994700193405151},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5345282554626465},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.500037670135498},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4992837905883789},{"id":"https://openalex.org/keywords/code","display_name":"Code (set theory)","score":0.49596819281578064},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.4889775514602661},{"id":"https://openalex.org/keywords/group-code","display_name":"Group code","score":0.45351025462150574},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.4498143792152405},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.43230539560317993},{"id":"https://openalex.org/keywords/linear-code","display_name":"Linear code","score":0.3608276844024658},{"id":"https://openalex.org/keywords/block-code","display_name":"Block code","score":0.2824201285839081},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.12846052646636963},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1225513219833374},{"id":"https://openalex.org/keywords/decoding-methods","display_name":"Decoding methods","score":0.08654674887657166}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.756916880607605},{"id":"https://openalex.org/C2096984","wikidata":"https://www.wikidata.org/wiki/Q245464","display_name":"Cyclic code","level":5,"score":0.6160991191864014},{"id":"https://openalex.org/C200127275","wikidata":"https://www.wikidata.org/wiki/Q173853","display_name":"Quaternion","level":2,"score":0.5994700193405151},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5345282554626465},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.500037670135498},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4992837905883789},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.49596819281578064},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.4889775514602661},{"id":"https://openalex.org/C3427991","wikidata":"https://www.wikidata.org/wiki/Q651426","display_name":"Group code","level":5,"score":0.45351025462150574},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.4498143792152405},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.43230539560317993},{"id":"https://openalex.org/C2400350","wikidata":"https://www.wikidata.org/wiki/Q1752667","display_name":"Linear code","level":4,"score":0.3608276844024658},{"id":"https://openalex.org/C157125643","wikidata":"https://www.wikidata.org/wiki/Q884707","display_name":"Block code","level":3,"score":0.2824201285839081},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.12846052646636963},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1225513219833374},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.08654674887657166},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.0},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","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},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207160.2015.1069819","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2015.1069819","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3332386192","display_name":null,"funder_award_id":"11471255,61171082","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W1520389133","https://openalex.org/W1594873819","https://openalex.org/W2034401304","https://openalex.org/W2108964585","https://openalex.org/W2151359594","https://openalex.org/W2167697875","https://openalex.org/W2963945674"],"related_works":["https://openalex.org/W1177344880","https://openalex.org/W4248511311","https://openalex.org/W1967648774","https://openalex.org/W3042543385","https://openalex.org/W3097204850","https://openalex.org/W1524903565","https://openalex.org/W2161553934","https://openalex.org/W2350050571","https://openalex.org/W4214485061","https://openalex.org/W1978124453"],"abstract_inverted_index":{"Let":[0],"Fq":[1,50,118],"be":[2],"a":[3,12,62,93,112,125],"finite":[4,72],"field":[5,73,135],"of":[6,14,39,51,78,96,119,129,142],"cardinality":[7],"q":[8,10],"where":[9,30,109],"is":[11,32,68,92,111,124],"power":[13],"an":[15,133,137],"odd":[16],"prime":[17],"integer,":[18],"and":[19,34,54,75,81,104,122,156,173],"denote":[20],"the":[21,27,40,100,105,153,158],"generalized":[22],"quaternion":[23,47],"group":[24,41],"Qn":[25],"by":[26],"presentation:":[28],"Qn=\u27e8x,y\u2223xn=1,y2=xn/2,yxy\u22121=x\u22121\u27e9":[29],"n":[31,121],"even":[33],"satisfies":[35],"gcd(n,q)=1.":[36],"Left":[37],"ideals":[38],"algebra":[42],"FqQn":[43],"are":[44,180],"called":[45],"left":[46,57,66,90,163,169,176],"codes":[48,80,98,151],"over":[49,117,132,171,178],"length":[52,120,130],"2n,":[53],"abbreviated":[55],"as":[56],"Qn-codes.":[58],"In":[59],"this":[60],"paper,":[61],"system":[63],"theory":[64,74,77],"for":[65,149,161],"Qn-codes":[67],"developed":[69],"only":[70],"using":[71],"basic":[76],"cyclic":[79,83,115],"skew":[82,126],"codes.":[84],"First,":[85],"we":[86,145],"prove":[87],"that":[88],"any":[89,162],"Qn-code":[91,164],"direct":[94],"sum":[95],"concatenated":[97,154],"with":[99],"inner":[101],"code":[102,107,116,128,160],"Ai":[103,110],"outer":[106,150],"Ci,":[108],"minimal":[113],"self-reciprocal":[114],"Ci":[123],"constacyclic":[127],"2":[131],"extension":[134,138],"or":[136],"principal":[139],"ideal":[140],"ring":[141],"Fq.":[143],"Then":[144],"give":[146],"explicit":[147],"expressions":[148],"in":[152],"codes,":[155],"present":[157],"dual":[159],"precisely.":[165],"Moreover,":[166],"all":[167,174],"distinct":[168,175],"Q12-codes":[170],"F5":[172],"Q20-codes":[177],"F3":[179],"presented,":[181],"respectively.":[182]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-04-26T08:31:28.666265","created_date":"2025-10-10T00:00:00"}
