{"id":"https://openalex.org/W2023194920","doi":"https://doi.org/10.1137/080715548","title":"Breaking the $\\epsilon$-Soundness Bound of the Linearity Test over GF(2)","display_name":"Breaking the $\\epsilon$-Soundness Bound of the Linearity Test over GF(2)","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W2023194920","doi":"https://doi.org/10.1137/080715548","mag":"2023194920"},"language":"en","primary_location":{"id":"doi:10.1137/080715548","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080715548","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","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/A5109110331","display_name":"Tali Kaufman","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tali Kaufman","raw_affiliation_strings":["kaufmant@mit.edu and ningxie@csail.mit.edu#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"kaufmant@mit.edu and ningxie@csail.mit.edu#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044662173","display_name":"Simon Litsyn","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Simon Litsyn","raw_affiliation_strings":["litsyn@eng.tau.ac.il#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"litsyn@eng.tau.ac.il#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5037751722","display_name":"Ning Xie","orcid":"https://orcid.org/0000-0002-6443-8111"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ning Xie","raw_affiliation_strings":["CSAIL/MIT"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"CSAIL/MIT","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.779,"has_fulltext":false,"cited_by_count":29,"citation_normalized_percentile":{"value":0.93059715,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"39","issue":"5","first_page":"1988","last_page":"2003"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998000264167786,"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"}},{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9991000294685364,"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/T10237","display_name":"Cryptography and Data Security","score":0.9987999796867371,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5843337178230286},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5407779812812805},{"id":"https://openalex.org/keywords/coset","display_name":"Coset","score":0.5311475992202759},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.46237656474113464},{"id":"https://openalex.org/keywords/boolean-function","display_name":"Boolean function","score":0.4258878231048584},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10710027813911438}],"concepts":[{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5843337178230286},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5407779812812805},{"id":"https://openalex.org/C85307737","wikidata":"https://www.wikidata.org/wiki/Q751969","display_name":"Coset","level":2,"score":0.5311475992202759},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.46237656474113464},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.4258878231048584},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10710027813911438}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/080715548","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080715548","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1398567088","display_name":"The Complexity of Testing Distributions","funder_award_id":"0514771","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320322252","display_name":"Israel Science Foundation","ror":"https://ror.org/04sazxf24"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":34,"referenced_works":["https://openalex.org/W104148001","https://openalex.org/W199049232","https://openalex.org/W1585475679","https://openalex.org/W1936868491","https://openalex.org/W1970630090","https://openalex.org/W1975810219","https://openalex.org/W1983510457","https://openalex.org/W1991006936","https://openalex.org/W1993625811","https://openalex.org/W1997855358","https://openalex.org/W1999032440","https://openalex.org/W2002864099","https://openalex.org/W2018925011","https://openalex.org/W2064334110","https://openalex.org/W2068634831","https://openalex.org/W2086653003","https://openalex.org/W2087964954","https://openalex.org/W2091921997","https://openalex.org/W2097206148","https://openalex.org/W2108860006","https://openalex.org/W2108900323","https://openalex.org/W2111385661","https://openalex.org/W2113121172","https://openalex.org/W2141815340","https://openalex.org/W2144461322","https://openalex.org/W2146261818","https://openalex.org/W2152221184","https://openalex.org/W2155600383","https://openalex.org/W2170716519","https://openalex.org/W2242669280","https://openalex.org/W2335400162","https://openalex.org/W2396166977","https://openalex.org/W3004537778","https://openalex.org/W4256317935"],"related_works":["https://openalex.org/W2388953866","https://openalex.org/W2349918401","https://openalex.org/W2052916897","https://openalex.org/W2362355910","https://openalex.org/W2126856880","https://openalex.org/W1979597421","https://openalex.org/W1501238516","https://openalex.org/W1956868877","https://openalex.org/W2066539480","https://openalex.org/W1523834128"],"abstract_inverted_index":{"For":[0],"Boolean":[1,57],"functions":[2],"that":[3,85,93,112,130],"are":[4,103],"$\\epsilon$-far":[5],"from":[6],"the":[7,14,18,25,46,116,121,149,157],"set":[8],"of":[9,24,56,124,152,156],"linear":[10],"functions,":[11],"we":[12,110,119],"study":[13],"lower":[15,122],"bound":[16,95,123],"on":[17,133,143],"rejection":[19],"probability":[20],"(denoted":[21],"by":[22,29,67,126],"$\\textsc{rej}(\\epsilon)$)":[23],"linearity":[26],"test":[27],"suggested":[28],"Blum,":[30],"Luby,":[31],"and":[32,49,148,165],"Rubinfeld":[33],"[J.":[34],"Comput.":[35],"System":[36],"Sci.,":[37],"47":[38],"(1993),":[39],"pp.":[40,77],"549\u2013595].":[41],"This":[42],"problem":[43,52],"is":[44,115,141],"arguably":[45],"most":[47],"fundamental":[48],"extensively":[50],"studied":[51],"in":[53],"property":[54],"testing":[55],"functions.":[58],"The":[59],"previously":[60],"best":[61],"bounds":[62],"for":[63,87,100,136],"$\\textsc{rej}(\\epsilon)$":[64,147],"were":[65],"obtained":[66],"Bellare":[68],"et":[69],"al.":[70],"[IEEE":[71],"Trans.":[72],"Inform.":[73],"Theory,":[74],"42":[75],"(1996),":[76],"1781\u20131795].":[78],"They":[79,90],"used":[80],"Fourier":[81,163],"analysis":[82,140,164],"to":[83,105,169],"show":[84,111],"$\\textsc{rej}(\\epsilon)\\geq\\epsilon$":[86,125],"every":[88,137],"$0\\leq\\epsilon\\leq1/2$.":[89,138],"also":[91],"conjectured":[92],"this":[94,108,113,171],"might":[96],"not":[97],"be":[98],"tight":[99],"$\\epsilon$'s":[101],"which":[102],"close":[104],"$1/2$.":[106],"In":[107],"paper":[109],"indeed":[114],"case.":[117],"Specifically,":[118],"improve":[120],"an":[127],"additive":[128],"constant":[129],"depends":[131],"only":[132],"$\\epsilon$:":[134],"$\\textsc{rej}(\\epsilon)\\geq\\epsilon+\\min\\{1376\\epsilon^{3}(1-2\\epsilon)^{12},\\frac{1}{4}\\epsilon(1-2\\epsilon)^{4}\\}$,":[135],"Our":[139],"based":[142],"a":[144,153],"relationship":[145],"between":[146],"weight":[150,172],"distribution":[151],"coset":[154],"code":[155],"Hadamard":[158],"code.":[159],"We":[160],"use":[161],"both":[162],"coding":[166],"theory":[167],"tools":[168],"estimate":[170],"distribution.":[173]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":4}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
