{"id":"https://openalex.org/W2001572656","doi":"https://doi.org/10.1145/1358183.1358185","title":"Irreducible polynomials and barker sequences","display_name":"Irreducible polynomials and barker sequences","publication_year":2007,"publication_date":"2007-12-01","ids":{"openalex":"https://openalex.org/W2001572656","doi":"https://doi.org/10.1145/1358183.1358185","mag":"2001572656"},"language":"en","primary_location":{"id":"doi:10.1145/1358183.1358185","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1358183.1358185","pdf_url":null,"source":{"id":"https://openalex.org/S176185422","display_name":"ACM communications in computer algebra","issn_l":"1932-2232","issn":["1932-2232","1932-2240"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320725","host_organization_name":"SIGSAM","host_organization_lineage":["https://openalex.org/P4310320725"],"host_organization_lineage_names":["SIGSAM"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Communications in Computer Algebra","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/A5029261392","display_name":"Peter Borwein","orcid":null},"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"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Peter Borwein","raw_affiliation_strings":["Simon Fraser University, Burnaby, B.C., Canada","Simon Fraser University, Burnaby, B.C., CANADA#TAB#"],"affiliations":[{"raw_affiliation_string":"Simon Fraser University, Burnaby, B.C., Canada","institution_ids":["https://openalex.org/I18014758"]},{"raw_affiliation_string":"Simon Fraser University, Burnaby, B.C., CANADA#TAB#","institution_ids":["https://openalex.org/I18014758"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056874049","display_name":"Erich Kaltofen","orcid":"https://orcid.org/0000-0003-2739-3230"},"institutions":[{"id":"https://openalex.org/I137902535","display_name":"North Carolina State University","ror":"https://ror.org/04tj63d06","country_code":"US","type":"education","lineage":["https://openalex.org/I137902535"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Erich Kaltofen","raw_affiliation_strings":["North Carolina State University, Raleigh, North Carolina"],"affiliations":[{"raw_affiliation_string":"North Carolina State University, Raleigh, North Carolina","institution_ids":["https://openalex.org/I137902535"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5087041917","display_name":"Michael J. Mossinghoff","orcid":"https://orcid.org/0000-0002-7983-5427"},"institutions":[{"id":"https://openalex.org/I141720752","display_name":"Davidson College","ror":"https://ror.org/02f7k4z58","country_code":"US","type":"education","lineage":["https://openalex.org/I141720752"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Michael J. Mossinghoff","raw_affiliation_strings":["Davidson College, Davidson, North Carolina"],"affiliations":[{"raw_affiliation_string":"Davidson College, Davidson, North Carolina","institution_ids":["https://openalex.org/I141720752"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5029261392"],"corresponding_institution_ids":["https://openalex.org/I18014758"],"apc_list":null,"apc_paid":null,"fwci":0.9673,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.81547842,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"41","issue":"4","first_page":"118","last_page":"121"},"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9961000084877014,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"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":0.9887999892234802,"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/mathematics","display_name":"Mathematics","score":0.842232346534729},{"id":"https://openalex.org/keywords/irreducibility","display_name":"Irreducibility","score":0.8217905759811401},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7731170058250427},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.6594567894935608},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.616586446762085},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.5457897782325745},{"id":"https://openalex.org/keywords/modulo","display_name":"Modulo","score":0.5137942433357239},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5009446144104004},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4461369812488556},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4452628791332245},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.4344406723976135},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.16804686188697815}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.842232346534729},{"id":"https://openalex.org/C2776823524","wikidata":"https://www.wikidata.org/wiki/Q6073561","display_name":"Irreducibility","level":2,"score":0.8217905759811401},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7731170058250427},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.6594567894935608},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.616586446762085},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5457897782325745},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.5137942433357239},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5009446144104004},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4461369812488556},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4452628791332245},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4344406723976135},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.16804686188697815},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1145/1358183.1358185","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1358183.1358185","pdf_url":null,"source":{"id":"https://openalex.org/S176185422","display_name":"ACM communications in computer algebra","issn_l":"1932-2232","issn":["1932-2232","1932-2240"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320725","host_organization_name":"SIGSAM","host_organization_lineage":["https://openalex.org/P4310320725"],"host_organization_lineage_names":["SIGSAM"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Communications in Computer Algebra","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.615.5885","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.615.5885","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.sigsam.org/bulletin/articles/162/irreducible.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G261290792","display_name":null,"funder_award_id":"CCF-0514585","funder_id":"https://openalex.org/F4320337387","funder_display_name":"Division of Computing and Communication Foundations"}],"funders":[{"id":"https://openalex.org/F4320337387","display_name":"Division of Computing and Communication Foundations","ror":"https://ror.org/01mng8331"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W627691335","https://openalex.org/W775608285","https://openalex.org/W1548207992","https://openalex.org/W2012641766","https://openalex.org/W2029700997","https://openalex.org/W2059196145","https://openalex.org/W2065609889","https://openalex.org/W2066747825","https://openalex.org/W4243644698","https://openalex.org/W6632654815","https://openalex.org/W6667295140"],"related_works":["https://openalex.org/W1630580488","https://openalex.org/W2007170860","https://openalex.org/W2177966018","https://openalex.org/W4302423208","https://openalex.org/W2196131019","https://openalex.org/W2112041331","https://openalex.org/W3087194925","https://openalex.org/W2467357782","https://openalex.org/W3152691454","https://openalex.org/W4288118777"],"abstract_inverted_index":{"A":[0,117],"Barker":[1,48,95,133],"sequence":[2,6,96],"is":[3,35,43,67,113],"a":[4,7,11,24,26,83,88,105,127],"finite":[5],"o":[8],",":[9,14,155],"...,":[10],"n":[12,34,52,66],"-1":[13],"each":[15],"term":[16],"\u00b11,":[17],"for":[18,62,87,121,156],"which":[19],"every":[20,157],"sum":[21],"\u03a3":[22],"i":[23,25,27],"+k":[28],"with":[29],"0":[30],"&lt;":[31,33],"k":[32],"either":[36],"0,":[37],"1,":[38],"or":[39],"--":[40],"1.":[41],"It":[42],"widely":[44],"conjectured":[45],"that":[46,71,131,145],"no":[47],"sequences":[49,134],"of":[50,85,91,97,109,119,135],"length":[51,98,137],"&gt;":[53],"13":[54],"exist,":[55],"and":[56],"this":[57,73,122],"conjecture":[58],"has":[59],"been":[60],"verified":[61],"the":[63,75,146],"case":[64,74],"when":[65],"odd.":[68],"We":[69],"show":[70],"in":[72,78,148],"problem":[76],"can":[77],"fact":[79],"be":[80],"reduced":[81],"to":[82],"question":[84,149],"irreducibility":[86,120],"certain":[89],"family":[90,123],"univariate":[92],"polynomials:":[93],"No":[94],"2":[99],"m":[100,112],"+":[101],"1":[102],"exists":[103],"if":[104],"particular":[106],"integer":[107],"polynomial":[108],"degree":[110],"4":[111],"irreducible":[114],"over":[115],"Q.":[116],"proof":[118,130],"would":[124],"thus":[125],"provide":[126],"short,":[128],"alternative":[129],"long":[132],"odd":[136],"do":[138],"not":[139],"exist.":[140],"However,":[141],"we":[142],"also":[143],"prove":[144],"polynomials":[147],"are":[150],"always":[151],"reducible":[152],"modulo":[153],"p":[154,159],"prime":[158],".":[160]},"counts_by_year":[{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
