{"id":"https://openalex.org/W7165675099","doi":"https://doi.org/10.48550/arxiv.2606.23579","title":"Optimal Small Set Expanders and Their Codes","display_name":"Optimal Small Set Expanders and Their Codes","publication_year":2026,"publication_date":"2026-06-22","ids":{"openalex":"https://openalex.org/W7165675099","doi":"https://doi.org/10.48550/arxiv.2606.23579"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.23579","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.23579","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2606.23579","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5135403149","display_name":"Tristram Bogart","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bogart, Tristram","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5078625962","display_name":"Marcelo Fiori","orcid":"https://orcid.org/0000-0002-3732-1778"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fiori, Marcelo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5139151493","display_name":"Pedro Raigorodsky","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Raigorodsky, Pedro","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5139214446","display_name":"Mauricio Velasco","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Velasco, Mauricio","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"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/T11797","display_name":"graph theory and CDMA systems","score":0.8758000135421753,"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"}},"topics":[{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.8758000135421753,"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/T11130","display_name":"Coding theory and cryptography","score":0.0560000017285347,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.030400000512599945,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/bipartite-graph","display_name":"Bipartite graph","score":0.6471999883651733},{"id":"https://openalex.org/keywords/expander-graph","display_name":"Expander graph","score":0.5443999767303467},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4226999878883362},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.42100000381469727},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4041999876499176},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.33869999647140503}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7196000218391418},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.6471999883651733},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6104000210762024},{"id":"https://openalex.org/C154547637","wikidata":"https://www.wikidata.org/wiki/Q776602","display_name":"Expander graph","level":3,"score":0.5443999767303467},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5131000280380249},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4226999878883362},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.42100000381469727},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4041999876499176},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.33869999647140503},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.3222000002861023},{"id":"https://openalex.org/C2779145032","wikidata":"https://www.wikidata.org/wiki/Q191494","display_name":"Digraph","level":2,"score":0.3174999952316284},{"id":"https://openalex.org/C2777434295","wikidata":"https://www.wikidata.org/wiki/Q959831","display_name":"Girth (graph theory)","level":2,"score":0.3124000132083893},{"id":"https://openalex.org/C122818955","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independent set","level":3,"score":0.28929999470710754},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2849000096321106},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.2687000036239624},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.2547999918460846}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.23579","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.23579","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.23579","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.23579","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"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":{"A":[0],"left-regular":[1],"bipartite":[2],"graph":[3,31],"$G$":[4],"of":[5,16,19,57,76,78,80,92],"degree":[6],"$d$":[7],"is":[8,32],"called":[9],"a":[10,30],"$(t,\u03b1)$-small-set-expander":[11],"if":[12,37],"every":[13,61],"subset":[14],"$X$":[15],"left":[17],"vertices":[18],"size":[20,81],"at":[21,25],"most":[22],"$t$":[23],"has":[24],"least":[26],"$\u03b1|X|$":[27],"neighbors.":[28],"Such":[29],"an":[33,86],"optimal":[34,48,93],"small-set":[35,94],"expander":[36],"small":[38],"subsets":[39],"have":[40],"as":[41,44,85],"many":[42],"neighbors":[43,77],"possible.":[45],"We":[46,63],"characterize":[47],"expanders":[49,59,95],"combinatorially":[50],"via":[51],"girth":[52],"and":[53],"prove":[54,65],"the":[55,74,90],"existence":[56],"$s$-optimal":[58],"for":[60,100],"$s$.":[62],"also":[64],"that":[66],"$s$-optimality":[67],"yields":[68],"new":[69],"\"transfer\"":[70],"lower":[71],"bounds":[72],"on":[73],"number":[75],"sets":[79],"$h\\geq":[82],"s$.":[83],"Finally,":[84],"application,":[87],"we":[88],"discuss":[89],"use":[91],"in":[96,104],"building":[97],"good":[98],"codes":[99],"key":[101],"exchange":[102],"protocols":[103],"post-quantum":[105],"cryptography.":[106]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-06-24T00:00:00"}
