{"id":"https://openalex.org/W1591811623","doi":"https://doi.org/10.1109/tit.2015.2469298","title":"Linear and Nonlinear Binary Kernels of Polar Codes of Small Dimensions With Maximum Exponents","display_name":"Linear and Nonlinear Binary Kernels of Polar Codes of Small Dimensions With Maximum Exponents","publication_year":2015,"publication_date":"2015-08-18","ids":{"openalex":"https://openalex.org/W1591811623","doi":"https://doi.org/10.1109/tit.2015.2469298","mag":"1591811623"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2015.2469298","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2015.2469298","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/A5048471624","display_name":"Hsien-Ping Lin","orcid":"https://orcid.org/0000-0002-7244-3375"},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Hsien-Ping Lin","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA, USA","institution_ids":["https://openalex.org/I84218800"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100772114","display_name":"Shu Lin","orcid":"https://orcid.org/0000-0003-1401-7834"},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Shu Lin","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA, USA","institution_ids":["https://openalex.org/I84218800"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102938606","display_name":"Khaled Abdel-Ghaffar","orcid":"https://orcid.org/0000-0002-3128-0557"},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Khaled A. S. Abdel-Ghaffar","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA, USA","institution_ids":["https://openalex.org/I84218800"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I84218800"],"apc_list":null,"apc_paid":null,"fwci":8.927,"has_fulltext":false,"cited_by_count":57,"citation_normalized_percentile":{"value":0.97464652,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":99},"biblio":{"volume":"61","issue":"10","first_page":"5253","last_page":"5270"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"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"}},"topics":[{"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/T11321","display_name":"Error Correcting Code Techniques","score":0.9983000159263611,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T13779","display_name":"Cancer Mechanisms and Therapy","score":0.9894999861717224,"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/exponent","display_name":"Exponent","score":0.8609055280685425},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.6881457567214966},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.6800333857536316},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6383944153785706},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6050458550453186},{"id":"https://openalex.org/keywords/kernel-method","display_name":"Kernel method","score":0.44463470578193665},{"id":"https://openalex.org/keywords/binary-code","display_name":"Binary code","score":0.4360443353652954},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.40997540950775146},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3817981481552124},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3202436566352844},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.2965683043003082},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.14413157105445862},{"id":"https://openalex.org/keywords/support-vector-machine","display_name":"Support vector machine","score":0.11406150460243225},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.0681382417678833},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.06646299362182617},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.06412485241889954}],"concepts":[{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.8609055280685425},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.6881457567214966},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6800333857536316},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6383944153785706},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6050458550453186},{"id":"https://openalex.org/C122280245","wikidata":"https://www.wikidata.org/wiki/Q620622","display_name":"Kernel method","level":3,"score":0.44463470578193665},{"id":"https://openalex.org/C63435697","wikidata":"https://www.wikidata.org/wiki/Q864135","display_name":"Binary code","level":3,"score":0.4360443353652954},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.40997540950775146},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3817981481552124},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3202436566352844},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2965683043003082},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.14413157105445862},{"id":"https://openalex.org/C12267149","wikidata":"https://www.wikidata.org/wiki/Q282453","display_name":"Support vector machine","level":2,"score":0.11406150460243225},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0681382417678833},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.06646299362182617},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.06412485241889954},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2015.2469298","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2015.2469298","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":[],"awards":[{"id":"https://openalex.org/G449475772","display_name":null,"funder_award_id":"CCF-1015548","funder_id":"https://openalex.org/F4320337387","funder_display_name":"Division of Computing and Communication Foundations"}],"funders":[{"id":"https://openalex.org/F4320308204","display_name":"Northrop Grumman","ror":"https://ror.org/05kewds18"},{"id":"https://openalex.org/F4320337387","display_name":"Division of Computing and Communication Foundations","ror":"https://ror.org/01mng8331"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":36,"referenced_works":["https://openalex.org/W58239232","https://openalex.org/W109405981","https://openalex.org/W359323642","https://openalex.org/W1568008612","https://openalex.org/W1606480398","https://openalex.org/W1946204087","https://openalex.org/W1971123554","https://openalex.org/W1975096885","https://openalex.org/W1984960835","https://openalex.org/W2010238614","https://openalex.org/W2025204728","https://openalex.org/W2038681062","https://openalex.org/W2048210712","https://openalex.org/W2055534149","https://openalex.org/W2062582191","https://openalex.org/W2069064669","https://openalex.org/W2096894382","https://openalex.org/W2098245138","https://openalex.org/W2100230274","https://openalex.org/W2104558348","https://openalex.org/W2112069403","https://openalex.org/W2119461808","https://openalex.org/W2122007939","https://openalex.org/W2125761068","https://openalex.org/W2139784773","https://openalex.org/W2145712176","https://openalex.org/W2150498905","https://openalex.org/W2154746932","https://openalex.org/W2159741270","https://openalex.org/W2166363667","https://openalex.org/W4205483176","https://openalex.org/W4255713959","https://openalex.org/W4295555409","https://openalex.org/W6602325378","https://openalex.org/W6634068109","https://openalex.org/W6680597225"],"related_works":["https://openalex.org/W2134880816","https://openalex.org/W1973746459","https://openalex.org/W2089892314","https://openalex.org/W2111298487","https://openalex.org/W4308157887","https://openalex.org/W2095626363","https://openalex.org/W2169565408","https://openalex.org/W1603091392","https://openalex.org/W2121506664","https://openalex.org/W2127229869"],"abstract_inverted_index":{"Polar":[0],"codes":[1,34],"are":[2,62,66,196,223],"constructed":[3,95],"based":[4,36],"on":[5,32,37],"kernels":[6,56,65,75,96,106,198,211],"with":[7,144,181,199],"polarizing":[8],"properties.":[9],"The":[10,27,64,130,210],"performance":[11],"of":[12,21,24,30,41,51,57,104,107,121,153,204,206,212],"a":[13,38,81,141,178,207],"polar":[14,33],"code":[15],"is":[16,35,84,91,112,160,166,173,177],"characterized":[17],"asymptotically":[18],"in":[19,101],"terms":[20],"the":[22,94,102,115,122,134,151,154],"exponent":[23,45,117,145,152,183],"its":[25],"kernel.":[26,209],"pioneering":[28],"work":[29],"Ar\u0131kan":[31],"linear":[39,52,142,155,179,208],"kernel":[40,143,156,180],"dimension":[42,108,136,165,186],"two":[43],"and":[44,53,80,193],"0.5.":[46],"In":[47],"this":[48,163],"paper,":[49],"constructions":[50],"nonlinear":[54,105,169,197,220],"binary":[55],"dimensions":[58,189,214],"up":[59,184],"to":[60,86,185],"16":[61],"presented.":[63],"obtained":[67,76],"using":[68],"computer":[69,78,82],"search":[70,79],"or":[71,125,225],"by":[72,77,158],"shortening":[73],"longer":[74],"program":[83],"used":[85],"determine":[87],"their":[88],"exponents.":[89],"It":[90],"proved":[92],"that":[93,114,120,133,175,205,215],"have":[97,216],"maximum":[98,116,182,217],"exponents":[99,200],"except":[100],"case":[103],"12":[109],"where":[110],"it":[111,172],"demonstrated":[113],"either":[118],"equals":[119],"presented":[123],"construction":[124],"assumes":[126],"another":[127],"specified":[128],"value.":[129],"results":[131],"show":[132],"minimum":[135,164],"for":[137,168],"which":[138],"there":[139,176,195],"exists":[140],"greater":[146],"than":[147,202],"0.5,":[148],"i.e.,":[149],"exceeds":[150],"proposed":[157],"Ar\u0131kan,":[159],"15,":[161,192],"while":[162],"14":[167],"kernels.":[170],"Furthermore,":[171],"shown":[174],"11.":[187],"For":[188],"13,":[190],"14,":[191],"16,":[194],"larger":[201],"any":[203],"these":[213],"exponent,":[218],"although":[219],"over":[221],"GF(2),":[222],"\u21244-linear":[224],"\u21242\u21244-linear.":[226]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":10},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":3}],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2025-10-10T00:00:00"}
