{"id":"https://openalex.org/W3023752339","doi":"https://doi.org/10.1137/22m1495597","title":"Rigid Matrices from Rectangular PCPs","display_name":"Rigid Matrices from Rectangular PCPs","publication_year":2024,"publication_date":"2024-04-03","ids":{"openalex":"https://openalex.org/W3023752339","doi":"https://doi.org/10.1137/22m1495597","mag":"3023752339"},"language":"en","primary_location":{"id":"doi:10.1137/22m1495597","is_oa":false,"landing_page_url":"https://doi.org/10.1137/22m1495597","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/A5012510300","display_name":"Amey Bhangale","orcid":"https://orcid.org/0000-0002-3878-9241"},"institutions":[{"id":"https://openalex.org/I103635307","display_name":"University of California, Riverside","ror":"https://ror.org/03nawhv43","country_code":"US","type":"education","lineage":["https://openalex.org/I103635307"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Amey Bhangale","raw_affiliation_strings":["Department of Computer Science and Engineering, University of California, Riverside, CA 92521 USA"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Engineering, University of California, Riverside, CA 92521 USA","institution_ids":["https://openalex.org/I103635307"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019248097","display_name":"Prahladh Harsha","orcid":"https://orcid.org/0000-0002-2739-5642"},"institutions":[{"id":"https://openalex.org/I11947397","display_name":"Tata Institute of Fundamental Research","ror":"https://ror.org/03ht1xw27","country_code":"IN","type":"education","lineage":["https://openalex.org/I11947397","https://openalex.org/I2799351866","https://openalex.org/I3149292468"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Prahladh Harsha","raw_affiliation_strings":["School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai 400005, India"],"affiliations":[{"raw_affiliation_string":"School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai 400005, India","institution_ids":["https://openalex.org/I11947397"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039003212","display_name":"Orr Paradise","orcid":"https://orcid.org/0000-0001-8212-8558"},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Orr Paradise","raw_affiliation_strings":["Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720 USA"],"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720 USA","institution_ids":["https://openalex.org/I95457486"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5049425955","display_name":"Avishay Tal","orcid":"https://orcid.org/0000-0002-0375-6554"},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Avishay Tal","raw_affiliation_strings":["Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720 USA"],"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720 USA","institution_ids":["https://openalex.org/I95457486"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5012510300"],"corresponding_institution_ids":["https://openalex.org/I103635307"],"apc_list":null,"apc_paid":null,"fwci":0.7608,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.69517048,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"53","issue":"2","first_page":"480","last_page":"523"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"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":0.9998999834060669,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9983999729156494,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/mathematics","display_name":"Mathematics","score":0.7321534752845764},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6732885837554932},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.591636598110199},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.5911404490470886},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.5188692808151245},{"id":"https://openalex.org/keywords/randomness","display_name":"Randomness","score":0.46671292185783386},{"id":"https://openalex.org/keywords/transformation","display_name":"Transformation (genetics)","score":0.4608227014541626},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.452700138092041},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4326923191547394},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.41430121660232544},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09187215566635132}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7321534752845764},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6732885837554932},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.591636598110199},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.5911404490470886},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.5188692808151245},{"id":"https://openalex.org/C125112378","wikidata":"https://www.wikidata.org/wiki/Q176640","display_name":"Randomness","level":2,"score":0.46671292185783386},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.4608227014541626},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.452700138092041},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4326923191547394},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41430121660232544},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09187215566635132},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/22m1495597","is_oa":false,"landing_page_url":"https://doi.org/10.1137/22m1495597","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/G904619761","display_name":null,"funder_award_id":"12-R&D-TFR-5.01-0500","funder_id":"https://openalex.org/F4320320768","funder_display_name":"Department of Atomic Energy, Government of India"}],"funders":[{"id":"https://openalex.org/F4320320768","display_name":"Department of Atomic Energy, Government of India","ror":"https://ror.org/02m388s04"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":54,"referenced_works":["https://openalex.org/W184142732","https://openalex.org/W198177064","https://openalex.org/W1502275094","https://openalex.org/W1523265685","https://openalex.org/W1527197079","https://openalex.org/W1530008367","https://openalex.org/W1967703498","https://openalex.org/W1970278864","https://openalex.org/W2002919335","https://openalex.org/W2004130530","https://openalex.org/W2007309095","https://openalex.org/W2008671178","https://openalex.org/W2019578639","https://openalex.org/W2019807548","https://openalex.org/W2022381972","https://openalex.org/W2030202224","https://openalex.org/W2038691311","https://openalex.org/W2044199052","https://openalex.org/W2045717693","https://openalex.org/W2048910070","https://openalex.org/W2053086236","https://openalex.org/W2061707159","https://openalex.org/W2064269488","https://openalex.org/W2083820240","https://openalex.org/W2087900794","https://openalex.org/W2117427692","https://openalex.org/W2125632628","https://openalex.org/W2135393523","https://openalex.org/W2137442467","https://openalex.org/W2140797111","https://openalex.org/W2144345645","https://openalex.org/W2146261818","https://openalex.org/W2148352980","https://openalex.org/W2158747250","https://openalex.org/W2163137752","https://openalex.org/W2263246245","https://openalex.org/W2510425678","https://openalex.org/W2553219432","https://openalex.org/W2742203643","https://openalex.org/W2951429494","https://openalex.org/W2953014891","https://openalex.org/W2965353432","https://openalex.org/W2965628617","https://openalex.org/W2971503262","https://openalex.org/W3004077558","https://openalex.org/W3004537778","https://openalex.org/W3012796491","https://openalex.org/W3115470548","https://openalex.org/W3118907271","https://openalex.org/W3119413233","https://openalex.org/W3127217102","https://openalex.org/W3129128072","https://openalex.org/W4220805816","https://openalex.org/W4313227250"],"related_works":["https://openalex.org/W3034924094","https://openalex.org/W3094954546","https://openalex.org/W1488708774","https://openalex.org/W1982811510","https://openalex.org/W4391100477","https://openalex.org/W2808944898","https://openalex.org/W2402189625","https://openalex.org/W4327779705","https://openalex.org/W1513698804","https://openalex.org/W4310560702"],"abstract_inverted_index":{".We":[0],"introduce":[1],"a":[2,64,88,111,203],"variant":[3],"of":[4,20,43,91,148,159,166,176,212],"Probabilistically":[5],"Checkable":[6],"Proofs":[7],"(PCPs)":[8],"that":[9,55,69,117,139],"we":[10,67,104,215],"refer":[11],"to":[12,196],"as":[13,21,78],"rectangular":[14,160],"PCPs,":[15],"wherein":[16],"proofs":[17,70],"are":[18,56,80,140],"thought":[19],"square":[22],"matrices,":[23,79],"and":[24,46,60,86,93,181,202,209,221],"the":[25,30,41,47,50,106,172,218],"random":[26],"coins":[27],"used":[28],"by":[29],"verifier":[31],"can":[32],"be":[33],"partitioned":[34],"into":[35],"two":[36],"disjoint":[37],"sets,":[38],"one":[39],"determining":[40,49],"row":[42],"each":[44],"query":[45],"other":[48],"column.":[51],"We":[52,192],"construct":[53],"PCPs":[54,161],"efficient,":[57],"short,":[58],"smooth,":[59],"(almost)":[61],"rectangular.":[62],"As":[63],"key":[65],"application,":[66],"show":[68,194],"for":[71,123,217],"hard":[72],"languages":[73],"in":[74,101,137,171],"NTIME\\(\\,(2^n)\\),":[75],"when":[76],"viewed":[77],"rigid":[81,99],"infinitely":[82,124],"often.":[83],"This":[84,206],"strengthens":[85],"simplifies":[87],"recent":[89],"result":[90],"Alman":[92],"Chen":[94],"[FOCS,":[95],"2019]":[96],"constructing":[97],"explicit":[98],"matrices":[100,134,147],"FNP.":[102],"Namely,":[103],"prove":[105,216],"following":[107],"theorem:":[108],"There":[109],"is":[110,119],"constant":[112],"\\(\\delta":[113,141],"\\in":[114],"(0,1)\\)":[115],"such":[116],"there":[118],"an":[120,164],"FNP-machine":[121],"that,":[122],"many":[125],"\\(N\\),":[126],"on":[127],"input":[128],"\\(1^N\\)":[129],"outputs":[130],"\\(N":[131],"\\times":[132],"N\\)":[133],"with":[135,163],"entries":[136],"\\(\\mathbb{F}_2\\)":[138],"N^2\\)-far":[142],"(in":[143],"Hamming":[144],"distance)":[145],"from":[146],"rank":[149],"at":[150],"most":[151],"\\(2^{\\log":[152],"N/\\Omega":[153],"(\\log":[154],"\\log":[155],"N)}\\).":[156],"Our":[157],"construction":[158],"starts":[162],"analysis":[165],"how":[167,195],"randomness":[168],"yields":[169],"queries":[170],"Reed\u2013Muller-based":[173],"outer":[174,219],"PCP":[175,200,220],"Ben-Sasson,":[177],"Goldreich,":[178],"Harsha,":[179],"Sudan,":[180],"Vadhan":[182],"[SIAM":[183],"J.":[184],"Comput.,":[185],"36":[186],"(2006),":[187],"pp.":[188],"889\u2013974;":[189],"CCC,":[190],"2005].":[191],"then":[193],"preserve":[197],"rectangularity":[198],"under":[199],"composition":[201],"smoothness-inducing":[204],"transformation.":[205],"warrants":[207],"refined":[208],"stronger":[210],"notions":[211],"rectangularity,":[213],"which":[214],"its":[222],"transforms.Keywordscomplexity":[223],"theorymatrix":[224],"rigidityprobabilistically":[225],"checkable":[226],"proofsMSC":[227],"codes68Q1068Q1568Q17":[228]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}