{"id":"https://openalex.org/W65376805","doi":null,"title":"Regular structures of lines in complex spaces","display_name":"Regular structures of lines in complex spaces","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W65376805","mag":"65376805"},"language":"en","primary_location":{"id":"mag:65376805","doi":null,"is_oa":false,"landing_page_url":"http://summit.sfu.ca/system/files/iritems1/9188/etd4176.pdf","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null},"volume":null,"type":"dissertation","is_traditional":true,"indexed_in":[],"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/A5081560748","display_name":"Mahdad Khatirinejad Fard","orcid":null,"observed_orcids":[]},"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":"Mahdad Khatirinejad Fard","raw_affiliation_strings":["Simon Fraser University CANADA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Simon Fraser University CANADA","institution_ids":["https://openalex.org/I18014758"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5081560748"],"corresponding_institution_ids":["https://openalex.org/I18014758"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10020","display_name":"Quantum Information and Cryptography","score":0.24369999766349792,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T10020","display_name":"Quantum Information and Cryptography","score":0.24369999766349792,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T11130","display_name":"Coding theory and cryptography","score":0.18960000574588776,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.09390000253915787,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/equiangular-lines","display_name":"equiangular lines","score":0.9259999990463257},{"id":"https://openalex.org/keywords/mutually-unbiased-bases","display_name":"mutually unbiased bases","score":0.9940000176429749},{"id":"https://openalex.org/keywords/heisenberg-weyl-group","display_name":"Weyl-Heisenberg group","score":0.8399999737739563},{"id":"https://openalex.org/keywords/prime-power-dimensions","display_name":"prime power dimensions","score":0.8820000290870667},{"id":"https://openalex.org/keywords/nonexistence-results","display_name":"non-existence results","score":0.7929999828338623}],"concepts":[{"id":"https://openalex.org/C158587675","wikidata":"https://www.wikidata.org/wiki/Q17160413","display_name":"Mutually unbiased bases","level":3,"score":0.89566969871521},{"id":"https://openalex.org/C41779539","wikidata":"https://www.wikidata.org/wiki/Q2543717","display_name":"Equiangular polygon","level":3,"score":0.8067700862884521},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7118405103683472},{"id":"https://openalex.org/C174072685","wikidata":"https://www.wikidata.org/wiki/Q1667469","display_name":"Prime power","level":3,"score":0.48011088371276855},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.43553152680397034},{"id":"https://openalex.org/C196644772","wikidata":"https://www.wikidata.org/wiki/Q3884964","display_name":"Orbit (dynamics)","level":2,"score":0.4188937544822693},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3508906364440918},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34211140871047974},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.31825897097587585},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.31809574365615845},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2688780426979065},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1613796353340149},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"mag:65376805","doi":null,"is_oa":false,"landing_page_url":"http://summit.sfu.ca/system/files/iritems1/9188/etd4176.pdf","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":73,"referenced_works":["https://openalex.org/W96453002","https://openalex.org/W173145114","https://openalex.org/W178039687","https://openalex.org/W585473508","https://openalex.org/W1483489440","https://openalex.org/W1509167021","https://openalex.org/W1512840059","https://openalex.org/W1528740882","https://openalex.org/W1539726851","https://openalex.org/W1542496577","https://openalex.org/W1543806621","https://openalex.org/W1553440965","https://openalex.org/W1556265897","https://openalex.org/W1568296425","https://openalex.org/W1599335667","https://openalex.org/W1608469714","https://openalex.org/W1624182979","https://openalex.org/W1631356911","https://openalex.org/W1639957736","https://openalex.org/W1656253992","https://openalex.org/W1886182640","https://openalex.org/W1969891477","https://openalex.org/W1977809994","https://openalex.org/W1979500630","https://openalex.org/W1979972099","https://openalex.org/W1983735761","https://openalex.org/W1988646993","https://openalex.org/W1989397443","https://openalex.org/W1992643273","https://openalex.org/W1996415070","https://openalex.org/W2003583756","https://openalex.org/W2009497707","https://openalex.org/W2015049075","https://openalex.org/W2030369304","https://openalex.org/W2033437505","https://openalex.org/W2033772417","https://openalex.org/W2035469609","https://openalex.org/W2037387604","https://openalex.org/W2037406168","https://openalex.org/W2046938896","https://openalex.org/W2047595535","https://openalex.org/W2049467095","https://openalex.org/W2050482406","https://openalex.org/W2050507755","https://openalex.org/W2051772613","https://openalex.org/W2054884346","https://openalex.org/W2057906938","https://openalex.org/W2086869478","https://openalex.org/W2087377426","https://openalex.org/W2094235750","https://openalex.org/W2094366212","https://openalex.org/W2099101264","https://openalex.org/W2100395640","https://openalex.org/W2104795888","https://openalex.org/W2121743251","https://openalex.org/W2131730048","https://openalex.org/W2133866430","https://openalex.org/W2152345982","https://openalex.org/W2182304189","https://openalex.org/W2183758324","https://openalex.org/W2317628658","https://openalex.org/W2335187233","https://openalex.org/W2478188808","https://openalex.org/W2584020404","https://openalex.org/W2920061280","https://openalex.org/W2950091396","https://openalex.org/W2952181186","https://openalex.org/W2953637949","https://openalex.org/W2954945477","https://openalex.org/W2969357538","https://openalex.org/W3043060850","https://openalex.org/W3103274847","https://openalex.org/W3200817088"],"related_works":["https://openalex.org/W2951330406","https://openalex.org/W2043024709","https://openalex.org/W1539726851","https://openalex.org/W19542714","https://openalex.org/W7100308293","https://openalex.org/W7208780980","https://openalex.org/W2258179867","https://openalex.org/W2047595535","https://openalex.org/W3210887166","https://openalex.org/W3082982411"],"abstract_inverted_index":{"We":[0,31,127,162,196,240,267,321],"study":[1,163],"the":[2,48,59,65,79,132,141,172,182,254,289],"properties":[3],"of":[4,7,13,36,53,69,74,90,111,116,122,134,137,148,165,174,181,206,218,237,245,256,272,291,314],"regular":[5],"structures":[6,89,205],"lines,":[8,138],"such":[9,77,98,135,190],"as":[10,99],"equiangular":[11,57,114,123,219,235,312],"sets":[12,164,217,313],"lines":[14,54,91,117,124,315],"and":[15,28,41,106,125,178,203,251,275,318],"mutually":[16],"unbiased":[17],"bases":[18,76,85],"(MUBs)":[19],"in":[20,39,47,95,153,192,294,316,330],"a":[21,33,72,112,119,129,175,230,242,264],"general":[22],"setting":[23],"that":[24,44,78,170,249,258,288],"includes":[25],"real,":[26],"complex":[27,42,222,317],"quaternionic":[29,49,319],"spaces.":[30,320],"formulate":[32],"common":[34,120],"generalization":[35,121],"several":[37,96,277],"results":[38],"real":[40],"spaces":[43],"also":[45,197,322],"hold":[46],"space.":[50],"\r\nA":[51],"set":[52,68,115,136,236],"is":[55,64,71,86,118,155,224,233],"called":[56],"if":[58,226],"angle":[60,80],"between":[61,81,200],"each":[62],"pair":[63],"same.":[66],"A":[67,221],"MUBs":[70,152,169,191],"collection":[73],"orthonormal":[75],"vectors":[82,209,274,293,326],"from":[83],"different":[84],"constant.":[87],"Regular":[88],"have":[92,210],"been":[93,211],"studied":[94],"fields":[97],"digital":[100],"communication,":[101],"quantum":[102],"computing,":[103],"discrete":[104],"mathematics":[105],"analysis.":[107],"\r\nOur":[108],"new":[109,243,284],"concept":[110],"multipartite":[113],"MUBs.":[126],"prove":[128,276,287],"bound":[130],"on":[131],"size":[133],"which":[139],"generalizes":[140],"well-known":[142],"absolute":[143],"upper":[144],"bounds.":[145],"\r\nThe":[146],"existence":[147],"d":[149,166,333],"+":[150,167],"1":[151,168],"Cd":[154,331],"only":[156],"known":[157],"for":[158,310],"prime":[159,193,296],"power":[160,194],"dimensions.":[161,195],"are":[171],"union":[173],"standard":[176],"basis":[177],"an":[179,186,234],"orbit":[180,228],"Weyl-Heisenberg":[183,231],"group.":[184],"As":[185],"example,":[187,281],"we":[188,286,306],"construct":[189,215],"show":[198],"connections":[199],"spherical":[201],"2-designs":[202],"other":[204],"lines.":[207,220,239],"\r\nFiducial":[208],"widely":[212],"used":[213],"to":[214,262],"large":[216],"vector":[223],"fiducial":[225,246,265,273,292,325],"its":[227],"under":[229],"group":[232],"d2":[238],"give":[241,307],"characterization":[244,285],"vectors,":[247],"one":[248],"simplifies":[250],"significantly":[252],"reduces":[253],"number":[255],"equations":[257],"must":[259],"be":[260],"solved":[261],"find":[263,323],"vector.":[266],"consider":[268],"some":[269,308],"possible":[270],"classes":[271],"nonexistence":[278],"results.":[279],"For":[280],"using":[282],"our":[283],"construction":[290],"small":[295],"dimensions":[297],"by":[298],"Appleby":[299],"(2005)":[300],"essentially":[301],"does":[302],"not":[303],"generalize.":[304],"\r\nFinally,":[305],"methods":[309],"constructing":[311],"numerical":[324],"with":[327],"high":[328],"precision":[329],",":[332],"\u2264":[334],"21.":[335]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2016-06-24T00:00:00"}