{"id":"https://openalex.org/W1602540072","doi":"https://doi.org/10.37236/1491","title":"Some Pairwise Balanced Designs","display_name":"Some Pairwise Balanced Designs","publication_year":1999,"publication_date":"1999-10-23","ids":{"openalex":"https://openalex.org/W1602540072","doi":"https://doi.org/10.37236/1491","mag":"1602540072"},"language":"en","primary_location":{"id":"doi:10.37236/1491","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1491","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v7i1r13/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v7i1r13/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5047659402","display_name":"Malcolm Greig","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Malcolm Greig","raw_affiliation_strings":["Greig Consulting 207-170 East Fifth St. North Vancouver BC, Canada, V7L 4L4"],"affiliations":[{"raw_affiliation_string":"Greig Consulting 207-170 East Fifth St. North Vancouver BC, Canada, V7L 4L4","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5047659402"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.9197,"has_fulltext":true,"cited_by_count":10,"citation_normalized_percentile":{"value":0.72362182,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":96},"biblio":{"volume":"7","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9998000264167786,"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"}},{"id":"https://openalex.org/T11798","display_name":"Optimal Experimental Design Methods","score":0.9728999733924866,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8258206844329834},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8045332431793213},{"id":"https://openalex.org/keywords/basis","display_name":"Basis (linear algebra)","score":0.726814329624176},{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.6688770651817322},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.6096750497817993},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.5729780793190002},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.4702538251876831},{"id":"https://openalex.org/keywords/element","display_name":"Element (criminal law)","score":0.4304409921169281},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.42025288939476013},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10684129595756531},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07963678240776062},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.07772421836853027}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8258206844329834},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8045332431793213},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.726814329624176},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.6688770651817322},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.6096750497817993},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5729780793190002},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.4702538251876831},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.4304409921169281},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42025288939476013},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10684129595756531},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07963678240776062},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.07772421836853027},{"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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/1491","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1491","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v7i1r13/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/1491","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1491","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v7i1r13/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320313015","display_name":"Mount Saint Vincent University","ror":"https://ror.org/03g3p3b82"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1602540072.pdf","grobid_xml":"https://content.openalex.org/works/W1602540072.grobid-xml"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W28221538","https://openalex.org/W36124134","https://openalex.org/W143461864","https://openalex.org/W164275096","https://openalex.org/W1488445482","https://openalex.org/W1506575296","https://openalex.org/W1976198551","https://openalex.org/W1982036829","https://openalex.org/W1991114516","https://openalex.org/W2001496048","https://openalex.org/W2008248976","https://openalex.org/W2029376662","https://openalex.org/W2051507061","https://openalex.org/W2051907025","https://openalex.org/W2055010924","https://openalex.org/W2058148091","https://openalex.org/W2060812201","https://openalex.org/W2069011336","https://openalex.org/W2069903885","https://openalex.org/W2090362028","https://openalex.org/W2146626764","https://openalex.org/W2752462646","https://openalex.org/W2920061280","https://openalex.org/W3189990552","https://openalex.org/W4239488153","https://openalex.org/W4243644698","https://openalex.org/W4249151119","https://openalex.org/W6600475739","https://openalex.org/W6600618482","https://openalex.org/W6604588003","https://openalex.org/W6607447126","https://openalex.org/W6635066040"],"related_works":["https://openalex.org/W2487162673","https://openalex.org/W2942366970","https://openalex.org/W2793211469","https://openalex.org/W2949152769","https://openalex.org/W4372354731","https://openalex.org/W2562400057","https://openalex.org/W1692008701","https://openalex.org/W2597588799","https://openalex.org/W4360593462","https://openalex.org/W2368346088"],"abstract_inverted_index":{"A":[0,45],"pairwise":[1],"balanced":[2],"design,":[3],"$B(K;v)$,":[4],"is":[5,34,48,86,90,98,131,139,151],"a":[6,26,30,42,49,91,99,111,132,140],"block":[7,13],"design":[8],"on":[9],"$v$":[10,39],"points,":[11],"with":[12,19,65],"sizes":[14],"taken":[15],"from":[16],"$K$,":[17,32],"and":[18,61,104,125,145,156,177],"every":[20],"pair":[21],"of":[22,37,70,80,101,142,163,174,188],"points":[23],"occurring":[24],"in":[25,184],"unique":[27],"block;":[28],"for":[29,40,51,93,115,134,161],"fixed":[31],"$B(K)$":[33],"the":[35,52,67,78,180],"set":[36,100,141],"all":[38],"which":[41],"$B(K;v)$":[43],"exists.":[44],"set,":[46,53],"$S$,":[47],"PBD-basis":[50,82],"$T$,":[54],"if":[55],"$T=B(S)$.":[56],"Let":[57],"$N_{a(m)}=\\{n:n\\equiv":[58],"a\\bmod":[59],"m\\}$,":[60],"$N_{\\geq":[62],"m}=\\{n:n\\geq":[63],"m\\}$;":[64],"$Q$":[66],"corresponding":[68],"restriction":[69],"$N$":[71],"to":[72,153,179],"prime":[73],"powers.":[74],"This":[75],"paper":[76],"addresses":[77],"existence":[79],"three":[81],"sets.":[83],"1.":[84],"It":[85],"shown":[87,128],"that":[88,129],"$Q_{1(8)}$":[89],"basis":[92,114,133],"$N_{1(8)}\\setminus":[94],"E$,":[95],"where":[96,137],"$E$":[97],"5":[102],"definite":[103,144,155],"117":[105],"possible":[106,147,158,165],"exceptions.":[107,148,159],"2.":[108],"We":[109],"construct":[110],"78":[112],"element":[113],"$N_{1(8)}$":[116],"with,":[117],"at":[118],"most,":[119],"64":[120],"inessential":[121],"elements.":[122],"3.":[123],"Bennett":[124],"Zhu":[126],"have":[127],"$Q_{\\geq8}$":[130],"$N_{\\geq8}\\setminus":[135],"E'$,":[136],"$E'$":[138],"43":[143],"606":[146],"Their":[149],"result":[150],"improved":[152],"48":[154],"470":[157],"(Constructions":[160],"35":[162],"these":[164],"exceptions":[166],"are":[167],"known.)":[168],"Finally,":[169],"we":[170],"provide":[171],"brief":[172],"details":[173],"some":[175],"improvements":[176],"corrections":[178],"generating/exception":[181],"sets":[182],"published":[183],"The":[185],"CRC":[186],"Handbook":[187],"Combinatorial":[189],"Designs.":[190]},"counts_by_year":[{"year":2017,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}