{"id":"https://openalex.org/W3205682614","doi":"https://doi.org/10.1109/tit.2021.3120090","title":"Hexagonal Run-Length Zero Capacity Region\u2014Part I: Analytical Proofs","display_name":"Hexagonal Run-Length Zero Capacity Region\u2014Part I: Analytical Proofs","publication_year":2021,"publication_date":"2021-10-15","ids":{"openalex":"https://openalex.org/W3205682614","doi":"https://doi.org/10.1109/tit.2021.3120090","mag":"3205682614"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2021.3120090","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2021.3120090","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/A5049503829","display_name":"Spencer Congero","orcid":"https://orcid.org/0000-0002-3099-5884"},"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":true,"raw_author_name":"Spencer Congero","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, CA, USA"],"raw_orcid":"https://orcid.org/0000-0002-3099-5884","affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, CA, USA","institution_ids":["https://openalex.org/I36258959"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072111370","display_name":"K. Zeger","orcid":"https://orcid.org/0000-0001-6415-1447"},"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":"Kenneth Zeger","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, CA, USA"],"raw_orcid":"https://orcid.org/0000-0001-6415-1447","affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, CA, USA","institution_ids":["https://openalex.org/I36258959"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5049503829"],"corresponding_institution_ids":["https://openalex.org/I36258959"],"apc_list":null,"apc_paid":null,"fwci":0.1613,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.53267601,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"68","issue":"1","first_page":"130","last_page":"152"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12162","display_name":"Cellular Automata and Applications","score":0.9952999949455261,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T11522","display_name":"VLSI and FPGA Design Techniques","score":0.984000027179718,"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/notation","display_name":"Notation","score":0.6761366128921509},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.47537961602211},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.4627191424369812},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.4246533513069153},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.34888753294944763},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.33547163009643555},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.12572479248046875},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09980076551437378},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.05080139636993408}],"concepts":[{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.6761366128921509},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.47537961602211},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.4627191424369812},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.4246533513069153},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.34888753294944763},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33547163009643555},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.12572479248046875},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09980076551437378},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.05080139636993408},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2021.3120090","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2021.3120090","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":37,"referenced_works":["https://openalex.org/W1530042113","https://openalex.org/W1579764623","https://openalex.org/W1586667417","https://openalex.org/W1653035750","https://openalex.org/W1976205112","https://openalex.org/W1982939827","https://openalex.org/W1988575935","https://openalex.org/W1994768492","https://openalex.org/W2016271082","https://openalex.org/W2030859021","https://openalex.org/W2031392878","https://openalex.org/W2031521232","https://openalex.org/W2053015204","https://openalex.org/W2055211453","https://openalex.org/W2074670256","https://openalex.org/W2091521721","https://openalex.org/W2096903743","https://openalex.org/W2104557113","https://openalex.org/W2104667481","https://openalex.org/W2105353526","https://openalex.org/W2114437981","https://openalex.org/W2114648280","https://openalex.org/W2117673653","https://openalex.org/W2125069069","https://openalex.org/W2158305334","https://openalex.org/W2158517136","https://openalex.org/W2168380555","https://openalex.org/W2294250243","https://openalex.org/W2582449852","https://openalex.org/W2768089104","https://openalex.org/W2963340341","https://openalex.org/W2963624396","https://openalex.org/W2963814981","https://openalex.org/W3034779354","https://openalex.org/W3098499649","https://openalex.org/W3105502192","https://openalex.org/W3205757119"],"related_works":["https://openalex.org/W2808944898","https://openalex.org/W3196207352","https://openalex.org/W2951724202","https://openalex.org/W3084261076","https://openalex.org/W4387820098","https://openalex.org/W1996614773","https://openalex.org/W2576399385","https://openalex.org/W2963408011","https://openalex.org/W4236324517","https://openalex.org/W2184046398"],"abstract_inverted_index":{"The":[0,35,168],"zero":[1,53,148,213],"capacity":[2,51,146,211,263],"region":[3],"for":[4,16,43,149,155,312],"hexagonal":[5],"<inline-formula":[6,21,28,37,59,66,75,82,92,100,107,116,127,134,159,218,233,249,270,283,305,315],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[7,22,29,38,60,67,76,83,93,101,108,117,128,135,160,219,234,250,271,284,306,316],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">":[8,23,30,39,61,68,77,84,94,102,109,118,129,136,161,220,235,251,272,285,307,317],"<tex-math":[9,24,31,40,62,69,78,85,95,103,110,119,130,137,162,221,236,252,273,286,308,318],"notation=\"LaTeX\">$(d,k)$":[10,41],"</tex-math></inline-formula>":[11,26,33,42,64,72,80,89,97,105,112,123,132,139,166,227,242,258,277,290,310,322],"run-length":[12],"constraints":[13],"is":[14,52,113,147,174,212,264],"known":[15],"many,":[17],"but":[18],"not":[19],"all,":[20],"notation=\"LaTeX\">$d$":[25,111],"and":[27,115,125,154,244,260,292],"notation=\"LaTeX\">$k$":[32],".":[34,140,167,323],"pairs":[36],"which":[44],"it":[45],"has":[46],"been":[47],"unproven":[48],"whether":[49],"the":[50,145,210,262,297],"or":[54,106],"positive":[55,265],"consist":[56],"of:":[57],"(i)":[58,216],"notation=\"LaTeX\">$k=d+2$":[63],"when":[65,81,98,133,217,232,248,269,282,304],"notation=\"LaTeX\">$d\\ge":[70],"2$":[71],";":[73,90,124],"(ii)":[74,157,231,268],"notation=\"LaTeX\">$k=d+3$":[79],"notation=\"LaTeX\">$d":[86,120,163,253,274,287,319],"\\ge":[87,121,164,320],"1$":[88],"(iii)":[91,247,281],"notation=\"LaTeX\">$k=d+4$":[96,309],"either":[99],"notation=\"LaTeX\">$d=4$":[104,138],"odd":[114,314],"3$":[122,289],"(iv)":[126],"notation=\"LaTeX\">$k=d+5$":[131],"Here,":[141],"we":[142,191,203],"prove":[143,208],"that":[144,190,209,261],"all":[150],"of":[151,181,188,199],"case":[152,156,215,230,246,267,280,294],"(i),":[153],"whenever":[158],"7$":[165],"method":[169],"used":[170],"in":[171,214,229,245,266,279,293],"this":[172,200],"paper":[173],"to":[175,184,207],"reduce":[176],"an":[177],"infinite":[178],"search":[179],"space":[180],"valid":[182],"labelings":[183],"a":[185],"finite":[186],"set":[187],"configurations":[189],"exhaustively":[192],"examine":[193],"using":[194],"backtracking.":[195],"In":[196],"Part":[197],"II":[198],"two-part":[201],"series,":[202],"use":[204],"automated":[205],"procedures":[206],"notation=\"LaTeX\">$2":[222],"\\le":[223,225,238,240],"d":[224,239],"9$":[226],",":[228,243,259,278,291,311],"notation=\"LaTeX\">$3":[237],"11$":[241,321],"\\in":[254,275],"\\{":[255],"4,5,7,9":[256],"\\}$":[257],"\\{1,2\\}$":[276],"=":[288],"(iv).":[295],"Thus,":[296],"only":[298],"remaining":[299],"unknown":[300],"cases":[301],"are":[302],"now":[303],"any":[313]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
