{"id":"https://openalex.org/W7125234453","doi":"https://doi.org/10.48550/arxiv.2601.12427","title":"Counterexamples, Constructions, and Nonexistence Results for Optimal Ternary Cyclic Codes","display_name":"Counterexamples, Constructions, and Nonexistence Results for Optimal Ternary Cyclic Codes","publication_year":2026,"publication_date":"2026-01-18","ids":{"openalex":"https://openalex.org/W7125234453","doi":"https://doi.org/10.48550/arxiv.2601.12427"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2601.12427","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.12427","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2601.12427","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5123508868","display_name":"Jingjun Bao","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bao, Jingjun","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5123460893","display_name":"Hanlin Zou","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zou, Hanlin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9157000184059143,"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":0.9157000184059143,"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":0.056699998676776886,"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.0035000001080334187,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/counterexample","display_name":"Counterexample","score":0.7389000058174133},{"id":"https://openalex.org/keywords/block-code","display_name":"Block code","score":0.5271000266075134},{"id":"https://openalex.org/keywords/ternary-operation","display_name":"Ternary operation","score":0.5040000081062317},{"id":"https://openalex.org/keywords/linear-code","display_name":"Linear code","score":0.3959999978542328},{"id":"https://openalex.org/keywords/minimum-distance","display_name":"Minimum distance","score":0.34200000762939453},{"id":"https://openalex.org/keywords/expander-code","display_name":"Expander code","score":0.3028999865055084}],"concepts":[{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.7389000058174133},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6470999717712402},{"id":"https://openalex.org/C157125643","wikidata":"https://www.wikidata.org/wiki/Q884707","display_name":"Block code","level":3,"score":0.5271000266075134},{"id":"https://openalex.org/C64452783","wikidata":"https://www.wikidata.org/wiki/Q1524945","display_name":"Ternary operation","level":2,"score":0.5040000081062317},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48899999260902405},{"id":"https://openalex.org/C2400350","wikidata":"https://www.wikidata.org/wiki/Q1752667","display_name":"Linear code","level":4,"score":0.3959999978542328},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3677000105381012},{"id":"https://openalex.org/C3018063660","wikidata":"https://www.wikidata.org/wiki/Q6865448","display_name":"Minimum distance","level":2,"score":0.34200000762939453},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.31470000743865967},{"id":"https://openalex.org/C108509264","wikidata":"https://www.wikidata.org/wiki/Q5420758","display_name":"Expander code","level":5,"score":0.3028999865055084},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.28850001096725464},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.2808000147342682},{"id":"https://openalex.org/C2776194381","wikidata":"https://www.wikidata.org/wiki/Q3965271","display_name":"Subclass","level":3,"score":0.27959999442100525},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2712000012397766}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2601.12427","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.12427","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2601.12427","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.12427","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Cyclic":[0],"codes":[1,8,32,76,91,110,127],"are":[2],"an":[3],"important":[4],"subclass":[5],"of":[6,56,74,108,136],"linear":[7],"with":[9,111,129],"wide":[10],"applications":[11],"in":[12],"communication":[13],"systems":[14],"and":[15,22,38,59,70,96],"data":[16],"storage":[17],"systems.":[18],"In":[19,49],"2013,":[20],"Ding":[21],"Helleseth":[23],"presented":[24],"nine":[25],"open":[26,61],"problems":[27,41,62],"on":[28,125],"optimal":[29,75,109,126],"ternary":[30],"cyclic":[31,90],"$\\mathcal{C}_{(1,e)}$.":[33],"While":[34],"the":[35,39,54,57,65,83,89,130,134],"first":[36,66],"two":[37,72,105],"sixth":[40],"have":[42],"been":[43],"fully":[44],"solved,":[45],"others":[46],"remain":[47],"open.":[48],"this":[50],"paper,":[51],"we":[52,87,103,120],"advance":[53],"study":[55],"third":[58,84],"fourth":[60],"by":[63],"providing":[64],"counterexamples":[67],"to":[68],"both":[69],"constructing":[71],"families":[73,107],"under":[77],"certain":[78],"conditions,":[79],"thereby":[80],"partially":[81],"solving":[82],"problem.":[85],"Furthermore,":[86],"investigate":[88],"$\\mathcal{C}_{(1,e)}$":[92,128],"where":[93],"$e(3^h\\pm":[94],"1)\\equiv\\frac{3^m-a}{2}\\pmod{3^m-1}$":[95],"$a$":[97],"is":[98],"odd.":[99],"For":[100,117],"$a\\equiv":[101,118],"3\\pmod{4}$,":[102],"present":[104],"new":[106],"parameters":[112,132],"$[3^m-1,3^m-1-2m,4]$,":[113],"generalizing":[114],"known":[115],"constructions.":[116],"1\\pmod{4}$,":[119],"obtain":[121],"several":[122],"nonexistence":[123],"results":[124],"aforementioned":[131],"revealing":[133],"constraints":[135],"such":[137],"codes.":[138]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-01-22T00:00:00"}
