{"id":"https://openalex.org/W2893602491","doi":"https://doi.org/10.1504/ijicot.2018.10016444","title":"Self-dual codes over Z&lt;SUB align=\"right\"&gt;4[&lt;i&gt;x&lt;/i&gt;]/(&lt;i&gt;x&lt;/i&gt;&lt;SUP align=\"right\"&gt;2&lt;/SUP&gt; + 2&lt;i&gt;x&lt;/i&gt;) and the Z&lt;SUB align=\"right\"&gt;4-images","display_name":"Self-dual codes over Z&lt;SUB align=\"right\"&gt;4[&lt;i&gt;x&lt;/i&gt;]/(&lt;i&gt;x&lt;/i&gt;&lt;SUP align=\"right\"&gt;2&lt;/SUP&gt; + 2&lt;i&gt;x&lt;/i&gt;) and the Z&lt;SUB align=\"right\"&gt;4-images","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2893602491","doi":"https://doi.org/10.1504/ijicot.2018.10016444","mag":"2893602491"},"language":"en","primary_location":{"id":"doi:10.1504/ijicot.2018.10016444","is_oa":false,"landing_page_url":"https://doi.org/10.1504/ijicot.2018.10016444","pdf_url":null,"source":{"id":"https://openalex.org/S185467048","display_name":"International Journal of Information and Coding Theory","issn_l":"1753-7703","issn":["1753-7703","1753-7711"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Information and Coding Theory","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/A5087561856","display_name":"Bahattin Y\u0131ld\u0131z","orcid":"https://orcid.org/0000-0001-8106-3123"},"institutions":[{"id":"https://openalex.org/I203172682","display_name":"Northern Arizona University","ror":"https://ror.org/0272j5188","country_code":"US","type":"education","lineage":["https://openalex.org/I203172682"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Bahattin Yildiz","raw_affiliation_strings":["Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86004, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86004, USA","institution_ids":["https://openalex.org/I203172682"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5070013407","display_name":"Abidin Kaya","orcid":"https://orcid.org/0000-0003-0175-1909"},"institutions":[{"id":"https://openalex.org/I4210165309","display_name":"Sampoerna University","ror":"https://ror.org/05t91sk49","country_code":"ID","type":"education","lineage":["https://openalex.org/I4210165309"]}],"countries":["ID"],"is_corresponding":false,"raw_author_name":"Abidin Kaya","raw_affiliation_strings":["Sampoerna Academy, L'Avenue Campus, 12780, Jakarta, Indonesia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Sampoerna Academy, L'Avenue Campus, 12780, Jakarta, Indonesia","institution_ids":["https://openalex.org/I4210165309"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.1076362,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"5","issue":"2","first_page":"142","last_page":"142"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9817000031471252,"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":0.9817000031471252,"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/T13779","display_name":"Cancer Mechanisms and Therapy","score":0.9542999863624573,"subfield":{"id":"https://openalex.org/subfields/2734","display_name":"Pathology and Forensic Medicine"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/circulant-matrix","display_name":"Circulant matrix","score":0.9102897644042969},{"id":"https://openalex.org/keywords/orthogonality","display_name":"Orthogonality","score":0.6851456165313721},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.655289888381958},{"id":"https://openalex.org/keywords/dual","display_name":"Dual (grammatical number)","score":0.6010665893554688},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5955302119255066},{"id":"https://openalex.org/keywords/ring","display_name":"Ring (chemistry)","score":0.48460516333580017},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4327346682548523},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13498470187187195},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.05398362874984741}],"concepts":[{"id":"https://openalex.org/C115973184","wikidata":"https://www.wikidata.org/wiki/Q245457","display_name":"Circulant matrix","level":2,"score":0.9102897644042969},{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.6851456165313721},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.655289888381958},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.6010665893554688},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5955302119255066},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.48460516333580017},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4327346682548523},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13498470187187195},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.05398362874984741},{"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/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1504/ijicot.2018.10016444","is_oa":false,"landing_page_url":"https://doi.org/10.1504/ijicot.2018.10016444","pdf_url":null,"source":{"id":"https://openalex.org/S185467048","display_name":"International Journal of Information and Coding Theory","issn_l":"1753-7703","issn":["1753-7703","1753-7711"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Information and Coding Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/11","score":0.6299999952316284,"display_name":"Sustainable cities and communities"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2139378013","https://openalex.org/W2560470131","https://openalex.org/W2049656902","https://openalex.org/W2394404719","https://openalex.org/W2903415888","https://openalex.org/W4287113494","https://openalex.org/W4292751759","https://openalex.org/W2293441232","https://openalex.org/W2964083059","https://openalex.org/W2964130583"],"abstract_inverted_index":{"In":[0],"this":[1],"work,":[2],"constructions":[3],"for":[4,52],"self-dual":[5,27,35,71],"codes":[6,36],"over":[7],"the":[8,32,42,53,93],"ring":[9],"Z4[x]/(x2":[10,57],"+":[11,58],"2x)":[12,59],"are":[13,60,65,90],"considered":[14],"together":[15],"with":[16],"their":[17],"images":[18],"under":[19],"an":[20],"orthogonality-preserving":[21],"grey":[22],"map,":[23],"which":[24,89],"result":[25],"in":[26],"Z4-codes.":[28],"Theoretical":[29],"results":[30],"about":[31],"existence/non-existence":[33],"of":[34,73],"from":[37],"construction":[38,63],"methods":[39,64],"such":[40],"as":[41],"double":[43,46],"circulant,":[44],"bordered":[45],"circulant":[47,50],"and":[48,56,78,85],"four":[49],"matrices":[51],"rings":[54],"Z4":[55],"given.":[61],"The":[62],"then":[66],"applied":[67],"to":[68],"get":[69],"good":[70],"Z4-codes":[72],"lengths":[74],"16,":[75],"32,":[76],"48":[77],"64":[79],"(including":[80],"some":[81],"extremal":[82],"type":[83,86],"I":[84],"II":[87],"codes),":[88],"tabulated":[91],"at":[92],"end.":[94]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
