{"id":"https://openalex.org/W2943019532","doi":"https://doi.org/10.1109/tit.2021.3136440","title":"Polar Codes for the Deletion Channel: Weak and Strong Polarization","display_name":"Polar Codes for the Deletion Channel: Weak and Strong Polarization","publication_year":2021,"publication_date":"2021-12-16","ids":{"openalex":"https://openalex.org/W2943019532","doi":"https://doi.org/10.1109/tit.2021.3136440","mag":"2943019532"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2021.3136440","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2021.3136440","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/A5075380828","display_name":"Ido Tal","orcid":"https://orcid.org/0000-0001-5419-1162"},"institutions":[{"id":"https://openalex.org/I174306211","display_name":"Technion \u2013 Israel Institute of Technology","ror":"https://ror.org/03qryx823","country_code":"IL","type":"education","lineage":["https://openalex.org/I174306211"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Ido Tal","raw_affiliation_strings":["Viterbi Department of Electrical and Computer Engineering, Technion, Haifa, Israel"],"raw_orcid":"https://orcid.org/0000-0001-5419-1162","affiliations":[{"raw_affiliation_string":"Viterbi Department of Electrical and Computer Engineering, Technion, Haifa, Israel","institution_ids":["https://openalex.org/I174306211"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034044119","display_name":"Henry D. Pfister","orcid":"https://orcid.org/0000-0001-5521-4397"},"institutions":[{"id":"https://openalex.org/I170897317","display_name":"Duke University","ror":"https://ror.org/00py81415","country_code":"US","type":"education","lineage":["https://openalex.org/I170897317"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Henry D. Pfister","raw_affiliation_strings":["Duke University, Durham, NC, USA"],"raw_orcid":"https://orcid.org/0000-0001-5521-4397","affiliations":[{"raw_affiliation_string":"Duke University, Durham, NC, USA","institution_ids":["https://openalex.org/I170897317"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5015830368","display_name":"Arman Fazeli","orcid":"https://orcid.org/0000-0001-8321-3391"},"institutions":[{"id":"https://openalex.org/I4210153776","display_name":"Apple (United States)","ror":"https://ror.org/059hsda18","country_code":"US","type":"company","lineage":["https://openalex.org/I4210153776"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Arman Fazeli","raw_affiliation_strings":["Apple Inc., Los Altos, CA, USA"],"raw_orcid":"https://orcid.org/0000-0001-8321-3391","affiliations":[{"raw_affiliation_string":"Apple Inc., Los Altos, CA, USA","institution_ids":["https://openalex.org/I4210153776"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5004102644","display_name":"Alexander Vardy","orcid":"https://orcid.org/0000-0003-3303-9078"},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Alexander Vardy","raw_affiliation_strings":["University of California at San Diego, San Diego, CA, USA"],"raw_orcid":"https://orcid.org/0000-0003-3303-9078","affiliations":[{"raw_affiliation_string":"University of California at San Diego, San Diego, CA, USA","institution_ids":["https://openalex.org/I36258959"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5075380828"],"corresponding_institution_ids":["https://openalex.org/I174306211"],"apc_list":null,"apc_paid":null,"fwci":1.7614,"has_fulltext":false,"cited_by_count":25,"citation_normalized_percentile":{"value":0.84845092,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"68","issue":"4","first_page":"2239","last_page":"2265"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11321","display_name":"Error Correcting Code Techniques","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T11321","display_name":"Error Correcting Code Techniques","score":0.9998000264167786,"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":0.9998000264167786,"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/T10207","display_name":"Advanced biosensing and bioanalysis techniques","score":0.9980000257492065,"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/trellis","display_name":"Trellis (graph)","score":0.6234749555587769},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6093915700912476},{"id":"https://openalex.org/keywords/decoding-methods","display_name":"Decoding methods","score":0.5715998411178589},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5298212170600891},{"id":"https://openalex.org/keywords/code-word","display_name":"Code word","score":0.5179087519645691},{"id":"https://openalex.org/keywords/channel","display_name":"Channel (broadcasting)","score":0.43779370188713074},{"id":"https://openalex.org/keywords/polar-code","display_name":"Polar code","score":0.4255768060684204},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3659444749355316},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2648562788963318},{"id":"https://openalex.org/keywords/telecommunications","display_name":"Telecommunications","score":0.14824840426445007}],"concepts":[{"id":"https://openalex.org/C2781235587","wikidata":"https://www.wikidata.org/wiki/Q7838009","display_name":"Trellis (graph)","level":3,"score":0.6234749555587769},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6093915700912476},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.5715998411178589},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5298212170600891},{"id":"https://openalex.org/C153207627","wikidata":"https://www.wikidata.org/wiki/Q863873","display_name":"Code word","level":3,"score":0.5179087519645691},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.43779370188713074},{"id":"https://openalex.org/C2778396592","wikidata":"https://www.wikidata.org/wiki/Q7209069","display_name":"Polar code","level":3,"score":0.4255768060684204},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3659444749355316},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2648562788963318},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.14824840426445007}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2021.3136440","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2021.3136440","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/G3190180327","display_name":null,"funder_award_id":"2018218","funder_id":"https://openalex.org/F4320320950","funder_display_name":"United States-Israel Binational Science Foundation"},{"id":"https://openalex.org/G6498293071","display_name":null,"funder_award_id":"CCF-1405119","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7792488542","display_name":null,"funder_award_id":"CCF-1719139","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8686538604","display_name":null,"funder_award_id":"1718494","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320320950","display_name":"United States-Israel Binational Science Foundation","ror":"https://ror.org/00j8z2m73"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":48,"referenced_works":["https://openalex.org/W1551519020","https://openalex.org/W1596594029","https://openalex.org/W1946204087","https://openalex.org/W1970026646","https://openalex.org/W1992506348","https://openalex.org/W2034381439","https://openalex.org/W2055534149","https://openalex.org/W2074437394","https://openalex.org/W2098245138","https://openalex.org/W2106045225","https://openalex.org/W2110009858","https://openalex.org/W2125870772","https://openalex.org/W2127323309","https://openalex.org/W2131586609","https://openalex.org/W2139784773","https://openalex.org/W2140129757","https://openalex.org/W2145712176","https://openalex.org/W2150498905","https://openalex.org/W2167082268","https://openalex.org/W2169071224","https://openalex.org/W2175334662","https://openalex.org/W2207982381","https://openalex.org/W2221448779","https://openalex.org/W2338291070","https://openalex.org/W2568570574","https://openalex.org/W2781719001","https://openalex.org/W2783704865","https://openalex.org/W2887116822","https://openalex.org/W2898657024","https://openalex.org/W2899702797","https://openalex.org/W2901536587","https://openalex.org/W2921868595","https://openalex.org/W2923015260","https://openalex.org/W2950808597","https://openalex.org/W2962799183","https://openalex.org/W2976477178","https://openalex.org/W2989130563","https://openalex.org/W3106529392","https://openalex.org/W3112599309","https://openalex.org/W3198709941","https://openalex.org/W4205483176","https://openalex.org/W4211192810","https://openalex.org/W4233413206","https://openalex.org/W4285719527","https://openalex.org/W6635681934","https://openalex.org/W6680597225","https://openalex.org/W6756520853","https://openalex.org/W6763943736"],"related_works":["https://openalex.org/W3009157344","https://openalex.org/W2008803526","https://openalex.org/W2090054304","https://openalex.org/W98604564","https://openalex.org/W2047078163","https://openalex.org/W287203899","https://openalex.org/W2786545028","https://openalex.org/W2885004924","https://openalex.org/W4386259870","https://openalex.org/W3081322325"],"abstract_inverted_index":{"This":[0,111],"paper":[1],"presents":[2],"the":[3,9,31,39,50,87,109,118,129,132,146,152,155],"first":[4],"proof":[5],"of":[6,26,102,108,124,131,154,164],"polarization":[7,80],"for":[8,82],"deletion":[10,15,32,88,147,156],"channel":[11,33,148,157],"with":[12,68],"a":[13,18,35,65,78,113,162],"constant":[14],"rate":[16,116],"and":[17,37,41,52,121],"regular":[19,165],"hidden-Markov":[20,166],"input":[21,167],"distribution.":[22],"A":[23],"key":[24],"part":[25],"this":[27,74,96,140],"work":[28],"involves":[29],"representing":[30],"using":[34],"trellis":[36,61,67],"describing":[38],"plus":[40,51],"minus":[42,53],"polar-decoding":[43],"operations":[44,54],"on":[45,86,145],"that":[46,139,151],"trellis.":[47],"In":[48],"particular,":[49],"can":[55,142,158],"be":[56,159],"seen":[57],"as":[58,70],"combining":[59],"adjacent":[60],"stages":[62],"to":[63],"yield":[64],"new":[66],"half":[69],"many":[71],"stages.":[72],"Using":[73],"viewpoint,":[75],"we":[76,94],"prove":[77],"weak":[79],"theorem":[81],"standard":[83],"polar":[84],"codes":[85],"channel.":[89],"To":[90],"achieve":[91,143],"strong":[92],"polarization,":[93],"modify":[95],"scheme":[97,114,141],"by":[98,137,149,161],"adding":[99],"guard":[100],"bands":[101],"repeated":[103],"zeros":[104],"between":[105],"various":[106],"parts":[107],"codeword.":[110],"gives":[112],"whose":[115,122],"approaches":[117],"mutual":[119],"information":[120],"probability":[123],"error":[125],"decays":[126],"exponentially":[127],"in":[128],"cube-root":[130],"block":[133],"length.":[134],"We":[135],"conclude":[136],"showing":[138],"capacity":[144,153],"proving":[150],"achieved":[160],"sequence":[163],"distributions.":[168]},"counts_by_year":[{"year":2025,"cited_by_count":10},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}