{"id":"https://openalex.org/W2771850059","doi":"https://doi.org/10.3934/amc.2017061","title":"Three basic questions on Boolean functions","display_name":"Three basic questions on Boolean functions","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2771850059","doi":"https://doi.org/10.3934/amc.2017061","mag":"2771850059"},"language":"en","primary_location":{"id":"doi:10.3934/amc.2017061","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2017061","pdf_url":null,"source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.3934/amc.2017061","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088687711","display_name":"Claude Carlet","orcid":"https://orcid.org/0000-0002-6118-7927"},"institutions":[{"id":"https://openalex.org/I4210102686","display_name":"Laboratoire Analyse, G\u00e9om\u00e9trie et Applications","ror":"https://ror.org/018nzqy79","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I4210091279","https://openalex.org/I4210102686","https://openalex.org/I4210141950","https://openalex.org/I48825208"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Claude Carlet","raw_affiliation_strings":["Laboratoire Analyse, G\u00e9om\u00e9trie et Applications"],"affiliations":[{"raw_affiliation_string":"Laboratoire Analyse, G\u00e9om\u00e9trie et Applications","institution_ids":["https://openalex.org/I4210102686"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5084453172","display_name":"Serge Feukoua","orcid":null},"institutions":[{"id":"https://openalex.org/I1331070479","display_name":"Universit\u00e9 de Yaound\u00e9 I","ror":"https://ror.org/022zbs961","country_code":"CM","type":"education","lineage":["https://openalex.org/I1331070479"]}],"countries":["CM"],"is_corresponding":false,"raw_author_name":"Serge Feukoua","raw_affiliation_strings":["University of Yaound\u00e9 1, Faculty of Sciences, Department of Mathematics, P.O.BOX 812 Yaound\u00e9, Cameroon"],"affiliations":[{"raw_affiliation_string":"University of Yaound\u00e9 1, Faculty of Sciences, Department of Mathematics, P.O.BOX 812 Yaound\u00e9, Cameroon","institution_ids":["https://openalex.org/I1331070479"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5088687711"],"corresponding_institution_ids":["https://openalex.org/I4210102686"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.18236371,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"11","issue":"4","first_page":"837","last_page":"855"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":1.0,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9950000047683716,"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/T13779","display_name":"Cancer Mechanisms and Therapy","score":0.9883999824523926,"subfield":{"id":"https://openalex.org/subfields/2734","display_name":"Pathology and Forensic Medicine"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8620470762252808},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.7404447793960571},{"id":"https://openalex.org/keywords/boolean-function","display_name":"Boolean function","score":0.6902865171432495},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6616522073745728},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5865575075149536},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5695139169692993},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.5568166375160217},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5533108115196228},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5442926287651062},{"id":"https://openalex.org/keywords/hyperplane","display_name":"Hyperplane","score":0.49950194358825684},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.47618189454078674},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.46141016483306885},{"id":"https://openalex.org/keywords/affine-transformation","display_name":"Affine transformation","score":0.458355575799942},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.18320274353027344}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8620470762252808},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.7404447793960571},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.6902865171432495},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6616522073745728},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5865575075149536},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5695139169692993},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.5568166375160217},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5533108115196228},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5442926287651062},{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.49950194358825684},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.47618189454078674},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46141016483306885},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.458355575799942},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.18320274353027344},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","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/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3934/amc.2017061","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2017061","pdf_url":null,"source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"},{"id":"pmh:oai:HAL:hal-03966912v1","is_oa":false,"landing_page_url":"https://hal.science/hal-03966912","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Advances in Mathematics of Communications, 2017, 11 (4), pp.837-855. ⟨10.3934/amc.2017061⟩","raw_type":"Journal articles"}],"best_oa_location":{"id":"doi:10.3934/amc.2017061","is_oa":true,"landing_page_url":"https://doi.org/10.3934/amc.2017061","pdf_url":null,"source":{"id":"https://openalex.org/S161492159","display_name":"Advances in Mathematics of Communications","issn_l":"1930-5338","issn":["1930-5338","1930-5346"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Advances in Mathematics of Communications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W1935580934","https://openalex.org/W2018946160","https://openalex.org/W2043200151","https://openalex.org/W2090432705","https://openalex.org/W2129120739","https://openalex.org/W2256579980","https://openalex.org/W2286164450","https://openalex.org/W2291741770","https://openalex.org/W2504274129","https://openalex.org/W2747869138","https://openalex.org/W6696932943"],"related_works":["https://openalex.org/W2369655046","https://openalex.org/W2019467317","https://openalex.org/W1585519779","https://openalex.org/W1965515989","https://openalex.org/W2375279953","https://openalex.org/W140666874","https://openalex.org/W1980053512","https://openalex.org/W2022655107","https://openalex.org/W2254112093","https://openalex.org/W4295628982"],"abstract_inverted_index":{"In":[0],"a":[1,147,174],"first":[2,90,100],"part":[3,176],"of":[4,49,86,153,163,177,184,237,245],"this":[5],"paper,":[6,179],"we":[7,25,52,82,180,211],"investigate":[8],"those":[9,27,54],"Boolean":[10,28,87],"functions":[11,29,55,154,215,230,247,262],"satisfying":[12],"two":[13,78],"apparently":[14],"related,":[15,81],"but":[16,102,114],"in":[17,207],"fact":[18],"distinct":[19],"conditions":[20],"concerning":[21],"the":[22,38,46,66,89,95,99,104,107,111,116,167,172,178,182,232,242,258],"algebraic":[23,40,47,69,169,249],"degree:1.":[24],"study":[26,53],"$f$":[30,50],"whose":[31,56,155,187],"restrictions":[32,156],"to":[33,43,74,157],"all":[34,65,129,158,201],"affine":[35,161],"hyperplanes":[36],"have":[37,64,166],"same":[39,67,168],"degree":[41,48,70,170,250],"(equal":[42],"$deg(f)$":[44],",":[45,60,63,132,134,140,146,192,199],"),2.":[51],"derivatives":[57,190],"$D_af(x)=f(x)+":[58],"f(x+a)$":[59],"$a\u2260":[61,193],"0$":[62],"(optimal)":[68],"$deg(f)-1$":[71],".For":[72],"determining":[73],"which":[75],"extent":[76],"these":[77],"questions":[79],"are":[80,200],"find":[83],"three":[84],"classes":[85],"functions:":[88],"class":[91,97,109,148],"satisfies":[92,98,110],"both":[93],"conditions,":[94],"second":[96,105,112,175,188],"condition":[101,113],"not":[103,115,221],"and":[106,127,143,210,240],"third":[108],"first.":[117],"We":[118,203,227,255],"also":[119],"give":[120],"for":[121,128,248,265],"any":[122],"fixed":[123],"positive":[124],"integer":[125],"$k$":[126,159],"integers":[130],"$n$":[131,219,266],"$p$":[133],"$s$":[135],"such":[136,246],"that":[137,213],"$p\u2265q":[138],"k+1$":[139,142],"$s\u2265q":[141],"$n\u2265q":[144],"ps$":[145],"(denoted":[149],"by":[150,231],"$C_{n,p,s}$":[151],")":[152],"-codimensional":[160],"subspaces":[162],"${\\Bbb":[164],"F}_2^n$":[165],"as":[171],"function.In":[173],"introduce":[181],"notion":[183],"second-order-bent":[185,214,229,261],"function,":[186],"order":[189],"$D_aD_bf(x)=f(x)+f(x+a)+f(x+b)+f(x+a+b)$":[191],"0,":[194,196],"b\u2260":[195],"a\u2260":[197],"b$":[198],"balanced.":[202],"exhibit":[204],"an":[205],"example":[206],"3":[208,224,251],"variables":[209],"prove":[212,241],"cannot":[216],"exist":[217,264],"if":[218],"is":[220],"congruent":[222],"with":[223],"mod":[225],"4.":[226],"characterize":[228],"Walsh":[233],"transform,":[234],"state":[235],"some":[236],"their":[238],"properties":[239],"non":[243],"existence":[244],"when":[252],"$n>3$":[253],".":[254,270],"leave":[256],"open":[257],"question":[259],"whether":[260],"can":[263],"larger":[267],"than":[268],"$3$":[269]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-03-25T13:04:00.132906","created_date":"2025-10-10T00:00:00"}