{"id":"https://openalex.org/W3036019187","doi":"https://doi.org/10.1109/tit.2020.3002993","title":"Vanishing Flats: A Combinatorial Viewpoint on the Planarity of Functions and Their Application","display_name":"Vanishing Flats: A Combinatorial Viewpoint on the Planarity of Functions and Their Application","publication_year":2020,"publication_date":"2020-06-17","ids":{"openalex":"https://openalex.org/W3036019187","doi":"https://doi.org/10.1109/tit.2020.3002993","mag":"3036019187"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2020.3002993","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2020.3002993","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/A5010168954","display_name":"Shuxing Li","orcid":"https://orcid.org/0000-0002-3405-3392"},"institutions":[{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]},{"id":"https://openalex.org/I95793202","display_name":"Otto-von-Guericke-Universit\u00e4t Magdeburg","ror":"https://ror.org/00ggpsq73","country_code":"DE","type":"education","lineage":["https://openalex.org/I95793202"]}],"countries":["CA","DE"],"is_corresponding":true,"raw_author_name":"Shuxing Li","raw_affiliation_strings":["Institute of Algebra and Geometry, Faculty of Mathematics, Otto von Guericke University Magdeburg, Universit\u00e4tsplatz 2, Magdeburg, Germany","Simon Fraser University, Burnaby, Canada"],"affiliations":[{"raw_affiliation_string":"Institute of Algebra and Geometry, Faculty of Mathematics, Otto von Guericke University Magdeburg, Universit\u00e4tsplatz 2, Magdeburg, Germany","institution_ids":["https://openalex.org/I95793202"]},{"raw_affiliation_string":"Simon Fraser University, Burnaby, Canada","institution_ids":["https://openalex.org/I18014758"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5045628286","display_name":"Wilfried Meidl","orcid":null},"institutions":[{"id":"https://openalex.org/I138211613","display_name":"Austrian Academy of Sciences","ror":"https://ror.org/03anc3s24","country_code":"AT","type":"government","lineage":["https://openalex.org/I138211613"]},{"id":"https://openalex.org/I4210158379","display_name":"Johann Radon Institute for Computational and Applied Mathematics","ror":"https://ror.org/05a94k872","country_code":"AT","type":"facility","lineage":["https://openalex.org/I138211613","https://openalex.org/I4210158379"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Wilfried Meidl","raw_affiliation_strings":["Austrian Academy of Sciences, Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria"],"affiliations":[{"raw_affiliation_string":"Austrian Academy of Sciences, Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria","institution_ids":["https://openalex.org/I4210158379","https://openalex.org/I138211613"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5078235960","display_name":"Alexandr Polujan","orcid":"https://orcid.org/0000-0002-0461-4793"},"institutions":[{"id":"https://openalex.org/I95793202","display_name":"Otto-von-Guericke-Universit\u00e4t Magdeburg","ror":"https://ror.org/00ggpsq73","country_code":"DE","type":"education","lineage":["https://openalex.org/I95793202"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Alexandr Polujan","raw_affiliation_strings":["Faculty of Mathematics, Institute of Algebra and Geometry, Otto von Guericke University Magdeburg, Magdeburg, Germany"],"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, Institute of Algebra and Geometry, Otto von Guericke University Magdeburg, Magdeburg, Germany","institution_ids":["https://openalex.org/I95793202"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013787536","display_name":"Alexander Pott","orcid":"https://orcid.org/0000-0002-1125-1003"},"institutions":[{"id":"https://openalex.org/I95793202","display_name":"Otto-von-Guericke-Universit\u00e4t Magdeburg","ror":"https://ror.org/00ggpsq73","country_code":"DE","type":"education","lineage":["https://openalex.org/I95793202"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Alexander Pott","raw_affiliation_strings":["Faculty of Mathematics, Institute of Algebra and Geometry, Otto von Guericke University Magdeburg, Magdeburg, Germany"],"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, Institute of Algebra and Geometry, Otto von Guericke University Magdeburg, Magdeburg, Germany","institution_ids":["https://openalex.org/I95793202"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5051580622","display_name":"Constanza Riera","orcid":null},"institutions":[{"id":"https://openalex.org/I179863766","display_name":"Western Norway University of Applied Sciences","ror":"https://ror.org/05phns765","country_code":"NO","type":"education","lineage":["https://openalex.org/I179863766"]}],"countries":["NO"],"is_corresponding":false,"raw_author_name":"Constanza Riera","raw_affiliation_strings":["Western Norway University of Applied Sciences, Bergen, Norway"],"affiliations":[{"raw_affiliation_string":"Western Norway University of Applied Sciences, Bergen, Norway","institution_ids":["https://openalex.org/I179863766"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5021290362","display_name":"Pantelimon St\u0103nic\u0103","orcid":"https://orcid.org/0000-0002-8622-7120"},"institutions":[{"id":"https://openalex.org/I35364215","display_name":"Naval Postgraduate School","ror":"https://ror.org/033yfkj90","country_code":"US","type":"education","lineage":["https://openalex.org/I1330347796","https://openalex.org/I3130687028","https://openalex.org/I35364215"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Pantelimon Stanica","raw_affiliation_strings":["Naval Postgraduate School, Monterey, CA, USA"],"affiliations":[{"raw_affiliation_string":"Naval Postgraduate School, Monterey, CA, USA","institution_ids":["https://openalex.org/I35364215"]}]}],"institutions":[],"countries_distinct_count":5,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5010168954"],"corresponding_institution_ids":["https://openalex.org/I18014758","https://openalex.org/I95793202"],"apc_list":null,"apc_paid":null,"fwci":0.9604,"has_fulltext":false,"cited_by_count":16,"citation_normalized_percentile":{"value":0.8092303,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"66","issue":"11","first_page":"7101","last_page":"7112"},"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":0.9994999766349792,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9965000152587891,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/planarity-testing","display_name":"Planarity testing","score":0.8253875970840454},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4893419146537781},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4024598002433777},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.36654701828956604},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.358910471200943},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3267766535282135}],"concepts":[{"id":"https://openalex.org/C68260962","wikidata":"https://www.wikidata.org/wiki/Q7200967","display_name":"Planarity testing","level":2,"score":0.8253875970840454},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4893419146537781},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4024598002433777},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.36654701828956604},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.358910471200943},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3267766535282135}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2020.3002993","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2020.3002993","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"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11","score":0.5}],"awards":[],"funders":[{"id":"https://openalex.org/F4320308269","display_name":"Alexander von Humboldt-Stiftung","ror":"https://ror.org/012kf4317"},{"id":"https://openalex.org/F4320310984","display_name":"Pacific Institute for the Mathematical Sciences","ror":"https://ror.org/04w49dw51"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W103329842","https://openalex.org/W1528065150","https://openalex.org/W1531334709","https://openalex.org/W1587687098","https://openalex.org/W1945897195","https://openalex.org/W1979658145","https://openalex.org/W1984812887","https://openalex.org/W2000881271","https://openalex.org/W2006630435","https://openalex.org/W2022609823","https://openalex.org/W2041328554","https://openalex.org/W2052178764","https://openalex.org/W2053266393","https://openalex.org/W2082143349","https://openalex.org/W2091168784","https://openalex.org/W2162542560","https://openalex.org/W2285925216","https://openalex.org/W2523624916","https://openalex.org/W2555021951","https://openalex.org/W2606905045","https://openalex.org/W2742188599","https://openalex.org/W2761692269","https://openalex.org/W2942941259","https://openalex.org/W2947669364","https://openalex.org/W4285719527","https://openalex.org/W6727265857"],"related_works":["https://openalex.org/W2982185603","https://openalex.org/W4288090301","https://openalex.org/W1978684236","https://openalex.org/W2963573247","https://openalex.org/W2164938550","https://openalex.org/W3003614666","https://openalex.org/W1816979946","https://openalex.org/W1598502370","https://openalex.org/W4297455769","https://openalex.org/W2380864953"],"abstract_inverted_index":{"For":[0],"a":[1,37,55,109,119,161],"function":[2],"$f$":[3,14,50,61,124],"from":[4],"$\\mathbb":[5,8],"{F}_{2}^{n}$":[6,9],"to":[7,157],",":[10],"the":[11,30,41,44,58,63,73,99,103,132,138,158,171],"planarity":[12],"of":[13,32,46,49,57,65,75,105,134,154,160],"is":[15],"usually":[16],"measured":[17],"by":[18,130],"its":[19],"differential":[20,23,84,87,94,100],"uniformity":[21,85],"and":[22,62,86,137,145,177],"spectrum.":[24,101],"In":[25,70,148],"this":[26,128],"paper,":[27],"we":[28,150],"propose":[29],"concept":[31,82],"vanishing":[33,47,76,106,135,155],"flats,":[34],"which":[35,89],"supplies":[36],"combinatorial":[38,110],"viewpoint":[39],"on":[40],"planarity.":[42],"First,":[43],"number":[45,74,133],"flats":[48,77,107,136,156],"can":[51],"be":[52],"regarded":[53],"as":[54,79],"measure":[56],"distance":[59],"between":[60,83],"set":[64,104],"almost":[66],"perfect":[67],"nonlinear":[68],"functions.":[69],"some":[71],"cases,":[72],"serves":[78],"an":[80,152],"\u201cintermediate\u201d":[81],"spectrum,":[88],"contains":[90],"more":[91],"information":[92,122],"than":[93,98],"uniformity,":[95],"however":[96],"less":[97],"Secondly,":[102],"forms":[108],"configuration":[111],"called":[112],"partial":[113,139],"quadruple":[114,140],"system,":[115],"since":[116],"it":[117],"conveys":[118],"detailed":[120],"structural":[121],"about":[123],".":[125],"We":[126,169],"initiate":[127],"study":[129],"considering":[131],"systems":[141],"associated":[142],"with":[143,173],"monomials":[144],"Dembowski-Ostrom":[146],"polynomials.":[147],"addition,":[149],"present":[151],"application":[153],"partition":[159],"vector":[162],"space":[163],"into":[164],"disjoint":[165],"equidimensional":[166],"affine":[167],"spaces.":[168],"conclude":[170],"paper":[172],"several":[174],"further":[175],"questions":[176],"challenges.":[178]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2}],"updated_date":"2026-03-27T14:29:43.386196","created_date":"2025-10-10T00:00:00"}