{"id":"https://openalex.org/W4415140812","doi":"https://doi.org/10.1007/s10623-026-01821-1","title":"Perfect codes in Cayley graphs of abelian groups","display_name":"Perfect codes in Cayley graphs of abelian groups","publication_year":2026,"publication_date":"2026-04-01","ids":{"openalex":"https://openalex.org/W4415140812","doi":"https://doi.org/10.1007/s10623-026-01821-1"},"language":"en","primary_location":{"id":"doi:10.1007/s10623-026-01821-1","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-026-01821-1","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-026-01821-1.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10623-026-01821-1.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048639170","display_name":"Peter J\u200e. Cameron","orcid":"https://orcid.org/0000-0003-3130-9505"},"institutions":[{"id":"https://openalex.org/I16835326","display_name":"University of St Andrews","ror":"https://ror.org/02wn5qz54","country_code":"GB","type":"education","lineage":["https://openalex.org/I16835326"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Peter J. Cameron","raw_affiliation_strings":["School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife, KY16 9SS, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife, KY16 9SS, UK","institution_ids":["https://openalex.org/I16835326"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5119982495","display_name":"Roro Sihui Yap","orcid":null},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"The University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Roro Sihui Yap","raw_affiliation_strings":["School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia","institution_ids":["https://openalex.org/I165779595"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5077967587","display_name":"Sanming Zhou","orcid":"https://orcid.org/0000-0001-9854-6076"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"The University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"Sanming Zhou","raw_affiliation_strings":["School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia","institution_ids":["https://openalex.org/I165779595"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5077967587"],"corresponding_institution_ids":["https://openalex.org/I165779595"],"apc_list":{"value":2890,"currency":"USD","value_usd":2890},"apc_paid":{"value":2890,"currency":"USD","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0127152,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"94","issue":"4","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":0.9923999905586243,"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":0.9923999905586243,"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":0.9918000102043152,"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/T12829","display_name":"Advanced Breast Cancer Therapies","score":0.9151999950408936,"subfield":{"id":"https://openalex.org/subfields/2740","display_name":"Pulmonary and Respiratory 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/cayley-graph","display_name":"Cayley graph","score":0.7300999760627747},{"id":"https://openalex.org/keywords/abelian-group","display_name":"Abelian group","score":0.6489999890327454},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6043000221252441},{"id":"https://openalex.org/keywords/trivially-perfect-graph","display_name":"Trivially perfect graph","score":0.44369998574256897},{"id":"https://openalex.org/keywords/hamming-bound","display_name":"Hamming bound","score":0.43309998512268066},{"id":"https://openalex.org/keywords/perfect-graph","display_name":"Perfect graph","score":0.3190000057220459}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7436000108718872},{"id":"https://openalex.org/C120204988","wikidata":"https://www.wikidata.org/wiki/Q859174","display_name":"Cayley graph","level":3,"score":0.7300999760627747},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7275999784469604},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.6489999890327454},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6043000221252441},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5254999995231628},{"id":"https://openalex.org/C170903109","wikidata":"https://www.wikidata.org/wiki/Q7844666","display_name":"Trivially perfect graph","level":5,"score":0.44369998574256897},{"id":"https://openalex.org/C166530166","wikidata":"https://www.wikidata.org/wiki/Q1970635","display_name":"Hamming bound","level":5,"score":0.43309998512268066},{"id":"https://openalex.org/C32972407","wikidata":"https://www.wikidata.org/wiki/Q1187620","display_name":"Perfect graph","level":5,"score":0.3190000057220459},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3149999976158142},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.2786000072956085},{"id":"https://openalex.org/C113203676","wikidata":"https://www.wikidata.org/wiki/Q220997","display_name":"Vertex-transitive graph","level":5,"score":0.27799999713897705},{"id":"https://openalex.org/C93446932","wikidata":"https://www.wikidata.org/wiki/Q1972803","display_name":"Perfect power","level":2,"score":0.2759000062942505},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.2621999979019165}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s10623-026-01821-1","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-026-01821-1","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-026-01821-1.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2507.11871","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2507.11871","pdf_url":"https://arxiv.org/pdf/2507.11871","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"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":null,"raw_type":"text"},{"id":"pmh:oai:arXiv.org:2507.11871","is_oa":true,"landing_page_url":"https://arxiv.org/abs/2507.11871","pdf_url":"https://arxiv.org/pdf/2507.11871","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2507.11871","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2507.11871","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":"Preprint"}],"best_oa_location":{"id":"doi:10.1007/s10623-026-01821-1","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-026-01821-1","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-026-01821-1.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3725389238","display_name":"Perfect codes in Cayley graphs","funder_award_id":"DP250104965","funder_id":"https://openalex.org/F4320334704","funder_display_name":"Australian Research Council"}],"funders":[{"id":"https://openalex.org/F4320320974","display_name":"University of Melbourne","ror":"https://ror.org/01ej9dk98"},{"id":"https://openalex.org/F4320334704","display_name":"Australian Research Council","ror":"https://ror.org/05mmh0f86"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4415140812.pdf","grobid_xml":"https://content.openalex.org/works/W4415140812.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"A":[1,62],"perfect":[2,64,107,111],"code":[3,65],"in":[4,34,41,59,66,96,113],"a":[5,24,74],"graph":[6],"$$\\Gamma":[7,67,84],"=":[8],"(V,":[9],"E)$$":[10],"<mml:math":[11,45,69,86],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[12,46,70,87],"<mml:mrow>":[13,47],"<mml:mi>\u0393</mml:mi>":[14,71,88],"<mml:mo>=</mml:mo>":[15],"<mml:mo>(</mml:mo>":[16],"<mml:mi>V</mml:mi>":[17,48],"<mml:mo>,</mml:mo>":[18],"<mml:mi>E</mml:mi>":[19],"<mml:mo>)</mml:mo>":[20],"</mml:mrow>":[21,51],"</mml:math>":[22,52,72,89],"is":[23,53,73,90],"subset":[25,75],"C":[26,35,60,76,97],"of":[27,77,83,116],"V":[28,78],"such":[29,79],"that":[30,80],"no":[31],"two":[32],"vertices":[33],"are":[36],"adjacent":[37,54,91],"and":[38,109],"every":[39,81],"vertex":[40,58,82,95],"$$V":[42],"\\setminus":[43],"C$$":[44],"<mml:mo>\\</mml:mo>":[49],"<mml:mi>C</mml:mi>":[50],"to":[55,92],"exactly":[56,93],"one":[57,94],".":[61,98],"total":[63,110],"$$":[68,85],"In":[99],"this":[100],"paper":[101],"we":[102],"prove":[103],"several":[104],"results":[105],"on":[106],"codes":[108,112],"Cayley":[114],"graphs":[115],"finite":[117],"abelian":[118],"groups.":[119]},"counts_by_year":[],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-14T00:00:00"}
