{"id":"https://openalex.org/W7133502964","doi":"https://doi.org/10.48550/arxiv.2603.03219","title":"Hardness of the Binary Covering Radius Problem in Large $\\ell_p$ Norms","display_name":"Hardness of the Binary Covering Radius Problem in Large $\\ell_p$ Norms","publication_year":2026,"publication_date":"2026-03-03","ids":{"openalex":"https://openalex.org/W7133502964","doi":"https://doi.org/10.48550/arxiv.2603.03219"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2603.03219","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.03219","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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.2603.03219","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5047830282","display_name":"Huck Bennett","orcid":"https://orcid.org/0000-0002-5469-8841"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bennett, Huck","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5128102869","display_name":"Peter Ly","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ly, Peter","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/T12404","display_name":"Mathematical Approximation and Integration","score":0.7491999864578247,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.7491999864578247,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T10237","display_name":"Cryptography and Data Security","score":0.09769999980926514,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.07800000160932541,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/norm","display_name":"Norm (philosophy)","score":0.697700023651123},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.6287000179290771},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.5726000070571899},{"id":"https://openalex.org/keywords/radius","display_name":"RADIUS","score":0.5307999849319458},{"id":"https://openalex.org/keywords/work","display_name":"Work (physics)","score":0.4505999982357025},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4456999897956848}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7515000104904175},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.697700023651123},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.6287000179290771},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5726000070571899},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.5307999849319458},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.4505999982357025},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4456999897956848},{"id":"https://openalex.org/C153658351","wikidata":"https://www.wikidata.org/wiki/Q746264","display_name":"Constant function","level":3,"score":0.4275999963283539},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.38269999623298645},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.3743000030517578},{"id":"https://openalex.org/C105462591","wikidata":"https://www.wikidata.org/wiki/Q3737844","display_name":"Binary function","level":3,"score":0.34619998931884766},{"id":"https://openalex.org/C112955886","wikidata":"https://www.wikidata.org/wiki/Q5656275","display_name":"Hardness of approximation","level":3,"score":0.34290000796318054},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3237000107765198},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2522999942302704}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2603.03219","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.03219","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2603.03219","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.03219","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[{"score":0.7250648736953735,"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0,138],"study":[1],"the":[2,5,14,60,112,125,147,162,176],"hardness":[3,62,103],"of":[4,63,71,83,109,142,164,171],"$\u03b3$-approximate":[6],"decisional":[7,126],"Covering":[8,114],"Radius":[9,115],"Problem":[10,116,129],"on":[11],"lattices":[12],"in":[13,131,134,146,175],"$\\ell_p$":[15],"norm":[16,133,149],"($\u03b3$-$\\text{GapCRP}_p$).":[17],"Specifically,":[18],"we":[19],"prove":[20],"that":[21,46],"there":[22],"is":[23,56],"an":[24,135],"explicit":[25,66],"function":[26],"$\u03b3(p)$,":[27],"with":[28],"$\u03b3(p)":[29],">":[30,34,51,154],"1$":[31],"for":[32,47,65,85,87,96,106,150],"$p":[33,67,97],"p_0":[35],"\\approx":[36],"35.31$":[37],"and":[38,73,77,95,124,159],"$\\lim_{p":[39],"\\to":[40],"\\infty}":[41],"\u03b3(p)":[42],"=":[43,98],"9/8$,":[44],"such":[45],"any":[48,132,151],"constant":[49,152],"$\\varepsilon":[50,153],"0$,":[52],"$(\u03b3(p)":[53],"-":[54,144,173],"\\varepsilon)$-$\\text{GapCRP}_p$":[55],"$\\mathsf{NP}$-hard.":[57],"This":[58],"shows":[59],"first":[61],"$\\text{GapCRP}_p$":[64,86],"<":[68],"\\infty$.":[69,99],"Work":[70],"Haviv":[72],"Regev":[74],"(CCC,":[75],"2006":[76],"CJTCS,":[78],"2012)":[79],"previously":[80],"showed":[81,169],"$\u03a0_2$-hardness":[82,141,170],"approximation":[84],"all":[88],"sufficiently":[89],"large":[90],"(but":[91],"non-explicit)":[92],"finite":[93],"$p$":[94],"In":[100],"fact,":[101],"our":[102],"results":[104],"hold":[105],"a":[107],"variant":[108],"$\\text{GapCRP}$":[110,123],"called":[111],"Binary":[113],"($\\text{BinGapCRP}$),":[117],"which":[118,168],"trivially":[119],"reduces":[120],"to":[121],"both":[122],"Linear":[127],"Discrepancy":[128],"($\\text{LinDisc}$)":[130],"approximation-preserving":[136],"way.":[137],"also":[139],"show":[140],"$(9/8":[143,172],"\\varepsilon)$-$\\text{BinGapCRP}$":[145],"$\\ell_{\\infty}$":[148,177],"0$.":[155],"Our":[156],"work":[157,163],"extends":[158],"heavily":[160],"uses":[161],"Manurangsi":[165],"(IPL,":[166],"2021),":[167],"\\varepsilon)$-$\\text{LinDisc}$":[174],"norm.":[178]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-03-05T00:00:00"}