{"id":"https://openalex.org/W1984184793","doi":"https://doi.org/10.1109/tit.2013.2270275","title":"Tight Bounds on Computing Error-Correcting Codes by Bounded-Depth Circuits With Arbitrary Gates","display_name":"Tight Bounds on Computing Error-Correcting Codes by Bounded-Depth Circuits With Arbitrary Gates","publication_year":2013,"publication_date":"2013-09-11","ids":{"openalex":"https://openalex.org/W1984184793","doi":"https://doi.org/10.1109/tit.2013.2270275","mag":"1984184793"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2013.2270275","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2013.2270275","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","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/A5078455094","display_name":"Anna G\u00e1l","orcid":"https://orcid.org/0000-0001-9772-3966"},"institutions":[{"id":"https://openalex.org/I86519309","display_name":"The University of Texas at Austin","ror":"https://ror.org/00hj54h04","country_code":"US","type":"education","lineage":["https://openalex.org/I86519309"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Anna Gal","raw_affiliation_strings":["Department of Computer Science, University of Texas, Austin, Austin, TX, USA","Department of Computer Science, University of Texas at Austin, Austin, TX, USA#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Texas, Austin, Austin, TX, USA","institution_ids":["https://openalex.org/I86519309"]},{"raw_affiliation_string":"Department of Computer Science, University of Texas at Austin, Austin, TX, USA#TAB#","institution_ids":["https://openalex.org/I86519309"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055834958","display_name":"Kristoffer Arnsfelt Hansen","orcid":"https://orcid.org/0000-0002-1155-8072"},"institutions":[{"id":"https://openalex.org/I204337017","display_name":"Aarhus University","ror":"https://ror.org/01aj84f44","country_code":"DK","type":"education","lineage":["https://openalex.org/I204337017"]}],"countries":["DK"],"is_corresponding":false,"raw_author_name":"Kristoffer Arnsfelt Hansen","raw_affiliation_strings":["Department of Computer Science, Aarhus University, Aarhus, Denmark"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Aarhus University, Aarhus, Denmark","institution_ids":["https://openalex.org/I204337017"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040922782","display_name":"Michal Kouck\u00fd","orcid":"https://orcid.org/0000-0003-0808-2269"},"institutions":[{"id":"https://openalex.org/I4210120266","display_name":"Czech Academy of Sciences, Institute of Mathematics","ror":"https://ror.org/02tv1yf50","country_code":"CZ","type":"facility","lineage":["https://openalex.org/I202391551","https://openalex.org/I4210120266"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Michal Koucky","raw_affiliation_strings":["Institute of Mathematics, Academy of Sciences, Prague, Czech Republic","Inst. of Math., Prague, Czech Republic"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, Academy of Sciences, Prague, Czech Republic","institution_ids":["https://openalex.org/I4210120266"]},{"raw_affiliation_string":"Inst. of Math., Prague, Czech Republic","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109848440","display_name":"Pavel Pudl\u00e1k","orcid":null},"institutions":[{"id":"https://openalex.org/I4210120266","display_name":"Czech Academy of Sciences, Institute of Mathematics","ror":"https://ror.org/02tv1yf50","country_code":"CZ","type":"facility","lineage":["https://openalex.org/I202391551","https://openalex.org/I4210120266"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Pavel Pudlak","raw_affiliation_strings":["Institute of Mathematics, Academy of Sciences, Prague, Czech Republic","Inst. of Math., Prague, Czech Republic"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, Academy of Sciences, Prague, Czech Republic","institution_ids":["https://openalex.org/I4210120266"]},{"raw_affiliation_string":"Inst. of Math., Prague, Czech Republic","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072846764","display_name":"Emanuele Viola","orcid":"https://orcid.org/0000-0001-6091-1824"},"institutions":[{"id":"https://openalex.org/I12912129","display_name":"Northeastern University","ror":"https://ror.org/04t5xt781","country_code":"US","type":"education","lineage":["https://openalex.org/I12912129"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Emanuele Viola","raw_affiliation_strings":["College of Computer and Information Science, Northeastern University, Boston, MA, USA","College of Computer and Information Science, Northeastern University , Boston, MA , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Computer and Information Science, Northeastern University, Boston, MA, USA","institution_ids":["https://openalex.org/I12912129"]},{"raw_affiliation_string":"College of Computer and Information Science, Northeastern University , Boston, MA , USA","institution_ids":["https://openalex.org/I12912129"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.5505,"has_fulltext":true,"cited_by_count":14,"citation_normalized_percentile":{"value":0.65194854,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"59","issue":"10","first_page":"6611","last_page":"6627"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12029","display_name":"DNA and Biological Computing","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},"topics":[{"id":"https://openalex.org/T12029","display_name":"DNA and Biological Computing","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9993000030517578,"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/T10237","display_name":"Cryptography and Data Security","score":0.998199999332428,"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/bounded-function","display_name":"Bounded function","score":0.70777428150177},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5714781880378723},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.5140796899795532},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.38618096709251404},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.35754847526550293},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.34094494581222534},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.33676183223724365}],"concepts":[{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.70777428150177},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5714781880378723},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.5140796899795532},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.38618096709251404},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35754847526550293},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.34094494581222534},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.33676183223724365},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1109/tit.2013.2270275","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2013.2270275","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.726.7558","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.726.7558","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.au.dk/%7Earnsfelt/Papers/ecc-circuits.pdf","raw_type":"text"},{"id":"pmh:oai:asep.lib.cas.cz:CavUnEpca/0422134","is_oa":false,"landing_page_url":"http://hdl.handle.net/11104/0228346","pdf_url":null,"source":{"id":"https://openalex.org/S7407055266","display_name":"ASEP","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:pure.atira.dk:publications/1ea71716-d939-405d-8294-f0f749d2cf1b","is_oa":false,"landing_page_url":"http://eccc.hpi-web.de/report/2011/150/","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""},{"id":"pmh:oai:pure.atira.dk:publications/3e821006-d439-4be0-890d-e8d932e89385","is_oa":false,"landing_page_url":"https://pure.au.dk/portal/en/publications/3e821006-d439-4be0-890d-e8d932e89385","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Gal, A, Hansen, K A, Koucky, M, Pudlak, P & Viola, E 2013, 'Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates', I E E E Transactions on Information Theory, vol. 59, no. 10, pp. 6611-6627. https://doi.org/10.1109/TIT.2013.2270275","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W1539074731","https://openalex.org/W1555946936","https://openalex.org/W1577741814","https://openalex.org/W1590864168","https://openalex.org/W1977292605","https://openalex.org/W1992906573","https://openalex.org/W2014482556","https://openalex.org/W2018047324","https://openalex.org/W2031713274","https://openalex.org/W2031934066","https://openalex.org/W2035532007","https://openalex.org/W2042194938","https://openalex.org/W2045064738","https://openalex.org/W2047342855","https://openalex.org/W2049815987","https://openalex.org/W2058264792","https://openalex.org/W2058859323","https://openalex.org/W2069788171","https://openalex.org/W2080295594","https://openalex.org/W2088559598","https://openalex.org/W2090190190","https://openalex.org/W2100661986","https://openalex.org/W2103897251","https://openalex.org/W2122270497","https://openalex.org/W2134490689","https://openalex.org/W2135393523","https://openalex.org/W2150212378","https://openalex.org/W2335996541","https://openalex.org/W2490907619","https://openalex.org/W3143506846","https://openalex.org/W6644303842","https://openalex.org/W6680864989"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W4409439182","https://openalex.org/W3013085049","https://openalex.org/W1560496689","https://openalex.org/W1992739770","https://openalex.org/W2053286651","https://openalex.org/W4386900933","https://openalex.org/W2181743346"],"abstract_inverted_index":{"We":[0],"bound":[1,83,89],"the":[2,61,70,85,112,144],"minimum":[3,15],"number$w$of":[4],"wires":[5],"needed":[6],"to":[7,59],"compute":[8],"any":[9,92],"(asymptotically":[10],"good)":[11],"error-correcting":[12],"code$C:\\{0,1\\}^{\\Omega":[13],"(n)}\\to\\{0,1\\}^{n}$with":[14],"distance$\\Omega":[16],"(n)$,":[17],"using":[18],"unbounded":[19],"fan-in":[20],"circuits":[21,147],"of":[22,114,146],"depth$d$with":[23],"arbitrary":[24],"gates.":[25],"Our":[26,134],"main":[27],"results":[28],"are:":[29],"1)":[30],"if$d=2$,":[31],"then$w=\\Theta":[32,39,47],"(n":[33,79],"({\\lg":[34,80],"n/\\lg\\lg":[35,81],"n})^{2})$;":[36],"2)":[37],"if$d=3$,":[38],"(n\\lg\\lg":[40],"n)$;":[41],"3)":[42],"if$d=2k$or$d=2k+1$for":[43],"some":[44],"integer$k\\geq":[45],"2$,":[46],"(n\\lambda_{k}(n))$,":[48],"where$\\lambda_{1}(n)=\\lceil\\lg":[49],"n\\rceil$,$\\lambda_{i+1}(n)=\\lambda_{i}^{\\ast}(n)$,":[50],"and":[51,72,160],"the$\\ast$operation":[52],"gives":[53,84],"how":[54],"many":[55],"times":[56],"one":[57],"has":[58],"iterate":[60],"function$\\lambda_{i}$to":[62],"reach":[63],"a":[64,100,140],"value":[65],"at":[66],"most":[67],"1":[68],"from":[69],"argument$n$;":[71],"4)":[73],"if$d=\\lg^{\\ast}n$,":[74],"then$w=O(n)$.":[75],"For":[76],"depth$d=2$,":[77],"our$\\Omega":[78],"n})^{2})$lower":[82],"largest":[86],"known":[87,119],"lower":[88,135],"for":[90,105,129],"computing":[91,130,148],"linear":[93],"map.":[94],"The":[95],"upper":[96],"bounds":[97,128,136],"imply":[98],"that":[99,143],"(necessarily":[101],"dense)":[102],"generator":[103],"matrix":[104],"our":[106],"code":[107],"can":[108],"be":[109],"written":[110],"as":[111],"product":[113],"two":[115],"sparse":[116],"matrices.":[117],"Using":[118],"techniques,":[120],"we":[121],"also":[122],"obtain":[123],"similar":[124],"(but":[125],"not":[126],"tight)":[127],"pairwise-independent":[131],"hash":[132],"functions.":[133],"are":[137],"based":[138],"on":[139],"superconcentrator-like":[141],"condition":[142,154],"graphs":[145],"good":[149],"codes":[150],"must":[151],"satisfy.":[152],"This":[153],"is":[155],"provably":[156],"intermediate":[157],"between":[158],"superconcentrators":[159],"their":[161],"weakenings":[162],"considered":[163],"before.":[164]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1}],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2025-10-10T00:00:00"}
