{"id":"https://openalex.org/W1975096885","doi":"https://doi.org/10.1109/tit.2014.2312181","title":"Source and Channel Polarization Over Finite Fields and Reed&amp;#x2013;Solomon Matrices","display_name":"Source and Channel Polarization Over Finite Fields and Reed&amp;#x2013;Solomon Matrices","publication_year":2014,"publication_date":"2014-03-18","ids":{"openalex":"https://openalex.org/W1975096885","doi":"https://doi.org/10.1109/tit.2014.2312181","mag":"1975096885"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2014.2312181","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2014.2312181","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/A5028668037","display_name":"Ryuhei Mori","orcid":"https://orcid.org/0000-0001-5474-5145"},"institutions":[{"id":"https://openalex.org/I114531698","display_name":"Tokyo Institute of Technology","ror":"https://ror.org/0112mx960","country_code":"JP","type":"education","lineage":["https://openalex.org/I114531698"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Ryuhei Mori","raw_affiliation_strings":["Department of Mathematical and Computing Science, Graduate School of Information Science and Engineering, Tokyo Institute of Technology, Tokyo, Japan","Dept. of Math. and Comput. Sci., Tokyo Inst. of Technology, Tokyo, Japan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical and Computing Science, Graduate School of Information Science and Engineering, Tokyo Institute of Technology, Tokyo, Japan","institution_ids":["https://openalex.org/I114531698"]},{"raw_affiliation_string":"Dept. of Math. and Comput. Sci., Tokyo Inst. of Technology, Tokyo, Japan","institution_ids":["https://openalex.org/I114531698"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5014391514","display_name":"Toshiyuki Tanaka","orcid":"https://orcid.org/0000-0001-5962-9119"},"institutions":[{"id":"https://openalex.org/I22299242","display_name":"Kyoto University","ror":"https://ror.org/02kpeqv85","country_code":"JP","type":"education","lineage":["https://openalex.org/I22299242"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Toshiyuki Tanaka","raw_affiliation_strings":["Department of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto, Japan","Dept. of Syst. Sci., Kyoto Univ., Kyoto, Japan"],"affiliations":[{"raw_affiliation_string":"Department of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto, Japan","institution_ids":["https://openalex.org/I22299242"]},{"raw_affiliation_string":"Dept. of Syst. Sci., Kyoto Univ., Kyoto, Japan","institution_ids":["https://openalex.org/I22299242"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5028668037"],"corresponding_institution_ids":["https://openalex.org/I114531698"],"apc_list":null,"apc_paid":null,"fwci":9.7598,"has_fulltext":false,"cited_by_count":79,"citation_normalized_percentile":{"value":0.98026635,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"60","issue":"5","first_page":"2720","last_page":"2736"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9995999932289124,"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/T12029","display_name":"DNA and Biological Computing","score":0.9983000159263611,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.8460966348648071},{"id":"https://openalex.org/keywords/polarization","display_name":"Polarization (electrochemistry)","score":0.6023478507995605},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.5573044419288635},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4976201355457306},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.44649070501327515},{"id":"https://openalex.org/keywords/prime-power","display_name":"Prime power","score":0.44462907314300537},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.43114325404167175},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4223243296146393},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.4042056202888489},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2896653711795807},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.16108116507530212}],"concepts":[{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.8460966348648071},{"id":"https://openalex.org/C205049153","wikidata":"https://www.wikidata.org/wiki/Q2698605","display_name":"Polarization (electrochemistry)","level":2,"score":0.6023478507995605},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.5573044419288635},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4976201355457306},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44649070501327515},{"id":"https://openalex.org/C174072685","wikidata":"https://www.wikidata.org/wiki/Q1667469","display_name":"Prime power","level":3,"score":0.44462907314300537},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.43114325404167175},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4223243296146393},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.4042056202888489},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2896653711795807},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.16108116507530212},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C147789679","wikidata":"https://www.wikidata.org/wiki/Q11372","display_name":"Physical chemistry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2014.2312181","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2014.2312181","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":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":31,"referenced_works":["https://openalex.org/W1542102056","https://openalex.org/W1606480398","https://openalex.org/W1825817408","https://openalex.org/W1984960835","https://openalex.org/W1986170574","https://openalex.org/W1989008275","https://openalex.org/W2055534149","https://openalex.org/W2065345097","https://openalex.org/W2093566764","https://openalex.org/W2096894382","https://openalex.org/W2098158374","https://openalex.org/W2099294566","https://openalex.org/W2104974061","https://openalex.org/W2105667172","https://openalex.org/W2129444266","https://openalex.org/W2139784773","https://openalex.org/W2142901448","https://openalex.org/W2145712176","https://openalex.org/W2150498905","https://openalex.org/W2155995390","https://openalex.org/W2158137498","https://openalex.org/W2160578128","https://openalex.org/W2161541170","https://openalex.org/W2162099497","https://openalex.org/W2166363667","https://openalex.org/W2172906094","https://openalex.org/W2295517275","https://openalex.org/W2952386913","https://openalex.org/W2964312190","https://openalex.org/W3037524284","https://openalex.org/W3099516920"],"related_works":["https://openalex.org/W2009566949","https://openalex.org/W2101989977","https://openalex.org/W2019668591","https://openalex.org/W4287205200","https://openalex.org/W1993953828","https://openalex.org/W3100852909","https://openalex.org/W4287871927","https://openalex.org/W2978220613","https://openalex.org/W3003622452","https://openalex.org/W2466745091"],"abstract_inverted_index":{"Polarization":[0],"phenomenon":[1],"over":[2,61],"any":[3,73],"finite":[4,63,100],"field":[5,44,64],"Fq":[6],"with":[7],"size":[8],"q":[9],"being":[10],"a":[11,14,21,42,53,59,62,126],"power":[12],"of":[13,23,27,58,82,85,111,129],"prime":[15,43],"is":[16,20,69,96,103,114],"considered.":[17],"This":[18],"problem":[19],"generalization":[22,128],"the":[24,33,80,83,88,107,118,130,138],"original":[25,139],"proposal":[26,140],"channel":[28,76],"polarization":[29,86],"by":[30,45,92,116,141],"Ar\u0131kan":[31,93],"for":[32,87],"binary":[34,89,134],"field,":[35],"as":[36,38,125],"well":[37],"its":[39],"extension":[40],"to":[41,98],"Sasoglu,":[46],"Telatar,":[47],"and":[48,55,75,94],"Ar\u0131kan.":[49,142],"In":[50],"this":[51],"paper,":[52],"necessary":[54],"sufficient":[56],"condition":[57],"matrix":[60,135],"F":[65],"<sub":[66],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[67],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">q</sub>":[68],"shown":[70,105],"under":[71],"which":[72,121],"source":[74],"are":[77],"polarized.":[78],"Furthermore,":[79],"result":[81],"speed":[84],"alphabet":[90],"obtained":[91],"Telatar":[95],"generalized":[97],"arbitrary":[99],"field.":[101],"It":[102],"also":[104],"that":[106],"asymptotic":[108],"error":[109],"probability":[110],"polar":[112],"codes":[113],"improved":[115],"using":[117],"Reed-Solomon":[119],"matrices,":[120],"can":[122],"be":[123],"regarded":[124],"natural":[127],"2":[131,133],"\u00d7":[132],"used":[136],"in":[137]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":11},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":10},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":7},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":11},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}