{"id":"https://openalex.org/W2296585621","doi":"https://doi.org/10.3390/e19090235","title":"Entropies of Weighted Sums in Cyclic Groups and an Application to Polar Codes","display_name":"Entropies of Weighted Sums in Cyclic Groups and an Application to Polar Codes","publication_year":2017,"publication_date":"2017-09-07","ids":{"openalex":"https://openalex.org/W2296585621","doi":"https://doi.org/10.3390/e19090235","mag":"2296585621"},"language":"en","primary_location":{"id":"doi:10.3390/e19090235","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e19090235","pdf_url":"https://www.mdpi.com/1099-4300/19/9/235/pdf?version=1504765877","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/19/9/235/pdf?version=1504765877","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Emmanuel Abbe","orcid":null},"institutions":[{"id":"https://openalex.org/I20089843","display_name":"Princeton University","ror":"https://ror.org/00hx57361","country_code":"US","type":"education","lineage":["https://openalex.org/I20089843"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Emmanuel Abbe","raw_affiliation_strings":["Department of Electrical Engineering, Princeton University, Princeton, NJ 08544-1000, USA","Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA"],"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Princeton University, Princeton, NJ 08544-1000, USA","institution_ids":["https://openalex.org/I20089843"]},{"raw_affiliation_string":"Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA","institution_ids":["https://openalex.org/I20089843"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Jiange Li","orcid":null},"institutions":[{"id":"https://openalex.org/I86501945","display_name":"University of Delaware","ror":"https://ror.org/01sbq1a82","country_code":"US","type":"education","lineage":["https://openalex.org/I86501945"]},{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jiange Li","raw_affiliation_strings":["Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA"],"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA","institution_ids":["https://openalex.org/I86501945"]}]},{"author_position":"last","author":{"id":null,"display_name":"Mokshay Madiman","orcid":"https://orcid.org/0000-0002-2992-1829"},"institutions":[{"id":"https://openalex.org/I86501945","display_name":"University of Delaware","ror":"https://ror.org/01sbq1a82","country_code":"US","type":"education","lineage":["https://openalex.org/I86501945"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mokshay Madiman","raw_affiliation_strings":["Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA","institution_ids":["https://openalex.org/I86501945"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I20089843"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.0948,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.79844732,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"19","issue":"9","first_page":"235","last_page":"235"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11321","display_name":"Error Correcting Code Techniques","score":0.5842999815940857,"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":0.5842999815940857,"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/T11130","display_name":"Coding theory and cryptography","score":0.22669999301433563,"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/T10964","display_name":"Wireless Communication Security Techniques","score":0.051899999380111694,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/polar","display_name":"Polar","score":0.5863000154495239},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5403000116348267},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.53329998254776},{"id":"https://openalex.org/keywords/generator-matrix","display_name":"Generator matrix","score":0.42899999022483826},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.4117000102996826},{"id":"https://openalex.org/keywords/generator","display_name":"Generator (circuit theory)","score":0.34150001406669617},{"id":"https://openalex.org/keywords/r\u00e9nyi-entropy","display_name":"R\u00e9nyi entropy","score":0.3393000066280365}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7784000039100647},{"id":"https://openalex.org/C29705727","wikidata":"https://www.wikidata.org/wiki/Q294562","display_name":"Polar","level":2,"score":0.5863000154495239},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5403000116348267},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.53329998254776},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4641999900341034},{"id":"https://openalex.org/C1842014","wikidata":"https://www.wikidata.org/wiki/Q1502046","display_name":"Generator matrix","level":3,"score":0.42899999022483826},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.4117000102996826},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40700000524520874},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.34150001406669617},{"id":"https://openalex.org/C142611142","wikidata":"https://www.wikidata.org/wiki/Q1433083","display_name":"R\u00e9nyi entropy","level":3,"score":0.3393000066280365},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.3336000144481659},{"id":"https://openalex.org/C92224547","wikidata":"https://www.wikidata.org/wiki/Q1673216","display_name":"Probability of error","level":2,"score":0.3246999979019165},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.311599999666214},{"id":"https://openalex.org/C52622258","wikidata":"https://www.wikidata.org/wiki/Q131222","display_name":"Information theory","level":2,"score":0.29269999265670776},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.272599995136261},{"id":"https://openalex.org/C44415725","wikidata":"https://www.wikidata.org/wiki/Q4913893","display_name":"Binary entropy function","level":3,"score":0.26759999990463257},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.26429998874664307},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.2522999942302704}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/e19090235","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e19090235","pdf_url":"https://www.mdpi.com/1099-4300/19/9/235/pdf?version=1504765877","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1512.00135","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1512.00135","pdf_url":"https://arxiv.org/pdf/1512.00135","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:doaj.org/article:74afc4fba18d4426a8340c398f0e1a30","is_oa":true,"landing_page_url":"https://doaj.org/article/74afc4fba18d4426a8340c398f0e1a30","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 19, Iss 9, p 235 (2017)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/19/9/235/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e19090235","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy; Volume 19; Issue 9; Pages: 235","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e19090235","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e19090235","pdf_url":"https://www.mdpi.com/1099-4300/19/9/235/pdf?version=1504765877","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1046991289","display_name":null,"funder_award_id":"CCF-1346564","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2894389027","display_name":null,"funder_award_id":"DMS-1409504 (CAREER)","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3045816707","display_name":null,"funder_award_id":"1319299","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3796015728","display_name":null,"funder_award_id":"CCF-1319299","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4468103044","display_name":null,"funder_award_id":"1346564","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5994546564","display_name":null,"funder_award_id":"1409504","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6671297155","display_name":null,"funder_award_id":"CAREER","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8357027711","display_name":null,"funder_award_id":"DMS-1409504","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","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"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2296585621.pdf","grobid_xml":"https://content.openalex.org/works/W2296585621.grobid-xml"},"referenced_works_count":50,"referenced_works":["https://openalex.org/W1629475227","https://openalex.org/W1973898585","https://openalex.org/W1999697648","https://openalex.org/W2008835981","https://openalex.org/W2012462504","https://openalex.org/W2020804212","https://openalex.org/W2031700647","https://openalex.org/W2041295460","https://openalex.org/W2045099620","https://openalex.org/W2055534149","https://openalex.org/W2069071813","https://openalex.org/W2077246889","https://openalex.org/W2100842176","https://openalex.org/W2102646476","https://openalex.org/W2107271148","https://openalex.org/W2111033825","https://openalex.org/W2118280520","https://openalex.org/W2131455012","https://openalex.org/W2133323343","https://openalex.org/W2135057628","https://openalex.org/W2138081888","https://openalex.org/W2145712176","https://openalex.org/W2150498905","https://openalex.org/W2153802318","https://openalex.org/W2158822560","https://openalex.org/W2161226162","https://openalex.org/W2165440928","https://openalex.org/W2166363667","https://openalex.org/W2166365531","https://openalex.org/W2169017678","https://openalex.org/W2169770534","https://openalex.org/W2170335255","https://openalex.org/W2172906094","https://openalex.org/W2185328402","https://openalex.org/W2217094038","https://openalex.org/W2325333723","https://openalex.org/W2326676472","https://openalex.org/W2336140998","https://openalex.org/W2531627753","https://openalex.org/W2763610741","https://openalex.org/W2895239415","https://openalex.org/W2912209775","https://openalex.org/W2951614950","https://openalex.org/W2951783141","https://openalex.org/W2963988569","https://openalex.org/W3037524284","https://openalex.org/W4214887932","https://openalex.org/W4233488604","https://openalex.org/W4239101296","https://openalex.org/W6682112832"],"related_works":[],"abstract_inverted_index":{"In":[0],"this":[1],"note,":[2],"the":[3,15,19,32,52,61,64,75,83,107,127],"following":[4],"basic":[5],"question":[6],"is":[7,66],"explored:":[8],"in":[9,90],"a":[10],"cyclic":[11],"group,":[12,34],"how":[13],"are":[14,79],"Shannon":[16],"entropies":[17],"of":[18,23,63,74,85,113],"sum":[20],"and":[21,110],"difference":[22,65],"i.i.d.":[24],"random":[25],"variables":[26],"related":[27,81],"to":[28,82,105,116,126],"each":[29],"other?":[30],"For":[31],"integer":[33],"we":[35],"show":[36],"that":[37,73],"they":[38],"can":[39],"differ":[40],"by":[41],"any":[42],"real":[43],"number":[44],"additively,":[45],"but":[46],"not":[47],"too":[48],"much":[49],"multiplicatively;":[50],"on":[51],"other":[53],"hand,":[54],"for":[55,98],"Z":[56,59],"/":[57],"3":[58],",":[60],"entropy":[62],"always":[67],"at":[68],"least":[69],"as":[70,72],"large":[71],"sum.":[76],"These":[77],"results":[78,115],"closely":[80],"study":[84],"more-sums-than-differences":[86],"(i.e.,":[87],"MSTD)":[88],"sets":[89],"additive":[91],"combinatorics.":[92],"We":[93],"also":[94],"investigate":[95],"polar":[96,118],"codes":[97,119],"q-ary":[99],"input":[100],"channels":[101],"using":[102],"non-canonical":[103],"kernels":[104],"construct":[106],"generator":[108],"matrix":[109],"present":[111],"applications":[112],"our":[114],"constructing":[117],"with":[120],"significantly":[121],"improved":[122],"error":[123],"probability":[124],"compared":[125],"canonical":[128],"construction.":[129]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2016-06-24T00:00:00"}
