{"id":"https://openalex.org/W1602646929","doi":"https://doi.org/10.37236/1464","title":"On the Structure and Classification of SOMAs: Generalizations of Mutually Orthogonal Latin Squares","display_name":"On the Structure and Classification of SOMAs: Generalizations of Mutually Orthogonal Latin Squares","publication_year":1999,"publication_date":"1999-07-04","ids":{"openalex":"https://openalex.org/W1602646929","doi":"https://doi.org/10.37236/1464","mag":"1602646929"},"language":"en","primary_location":{"id":"doi:10.37236/1464","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1464","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v6i1r32/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/v6i1r32/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5049800727","display_name":"Leonard H. Soicher","orcid":"https://orcid.org/0000-0001-5836-8914"},"institutions":[{"id":"https://openalex.org/I166337079","display_name":"Queen Mary University of London","ror":"https://ror.org/026zzn846","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I166337079"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Leonard H. Soicher","raw_affiliation_strings":["School of Mathematical Sciences Queen Mary and Westfield College Mile End Road, London E1 4NS, U.K","Queen Mary University of London"],"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences Queen Mary and Westfield College Mile End Road, London E1 4NS, U.K","institution_ids":[]},{"raw_affiliation_string":"Queen Mary University of London","institution_ids":["https://openalex.org/I166337079"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5049800727"],"corresponding_institution_ids":["https://openalex.org/I166337079"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":25,"citation_normalized_percentile":{"value":0.05874912,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"6","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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8502229452133179},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7262961268424988},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.6625339984893799},{"id":"https://openalex.org/keywords/soma","display_name":"Soma","score":0.6367981433868408},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.4222847819328308},{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.415262371301651},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.4131473898887634},{"id":"https://openalex.org/keywords/latin-square","display_name":"Latin square","score":0.4122391939163208},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3635447323322296},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.11099892854690552},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10900789499282837},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09319460391998291}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8502229452133179},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7262961268424988},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6625339984893799},{"id":"https://openalex.org/C2779617337","wikidata":"https://www.wikidata.org/wiki/Q842429","display_name":"Soma","level":2,"score":0.6367981433868408},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.4222847819328308},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.415262371301651},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.4131473898887634},{"id":"https://openalex.org/C158887663","wikidata":"https://www.wikidata.org/wiki/Q679367","display_name":"Latin square","level":4,"score":0.4122391939163208},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3635447323322296},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.11099892854690552},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10900789499282837},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09319460391998291},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C70262065","wikidata":"https://www.wikidata.org/wiki/Q427652","display_name":"Rumen","level":3,"score":0.0},{"id":"https://openalex.org/C31903555","wikidata":"https://www.wikidata.org/wiki/Q1637030","display_name":"Food science","level":1,"score":0.0},{"id":"https://openalex.org/C169760540","wikidata":"https://www.wikidata.org/wiki/Q207011","display_name":"Neuroscience","level":1,"score":0.0},{"id":"https://openalex.org/C100544194","wikidata":"https://www.wikidata.org/wiki/Q41760","display_name":"Fermentation","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/1464","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1464","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v6i1r32/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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.45.9119","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.9119","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/Volume_6/../Volume_6/PostScriptfiles/v6i1r32.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.46.6669","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.6669","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.sztaki.hu/mirrors/EMIS/journals/EJC/Volume_6/PDF/v6i1r32.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1464","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1464","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v6i1r32/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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1602646929.pdf","grobid_xml":"https://content.openalex.org/works/W1602646929.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W32262330","https://openalex.org/W56998983","https://openalex.org/W1971685893","https://openalex.org/W1977622438","https://openalex.org/W1985301313","https://openalex.org/W1989957326","https://openalex.org/W2064658813"],"related_works":["https://openalex.org/W2101180249","https://openalex.org/W2040277507","https://openalex.org/W2323108107","https://openalex.org/W2734892631","https://openalex.org/W2052627508","https://openalex.org/W2053428912","https://openalex.org/W2020053772","https://openalex.org/W2035433397","https://openalex.org/W2107514012","https://openalex.org/W1970142267"],"abstract_inverted_index":{"Let":[0],"$k\\ge0$":[1],"and":[2,39,47,80,125,131,139,145],"$n\\ge2$":[3],"be":[4,64,85],"integers.":[5],"A":[6,61],"SOMA,":[7],"or":[8],"more":[9,55],"specifically":[10],"a":[11,24,82,88,167,174],"SOMA$(k,n)$,":[12],"is":[13,52],"an":[14],"$n\\times":[15],"n$":[16],"array":[17],"$A$,":[18,46],"whose":[19],"entries":[20],"are":[21],"$k$-subsets":[22],"of":[23,31,45,50,59,73,90,96,107,121,133,159,170],"$kn$-set":[25],"$\\Omega$,":[26],"such":[27],"that":[28],"each":[29,37,43],"element":[30],"$\\Omega$":[32,51],"occurs":[33],"exactly":[34,40],"once":[35,41],"in":[36,42,54,128],"row":[38],"column":[44],"no":[48],"2-subset":[49],"contained":[53],"than":[56],"one":[57],"entry":[58],"$A$.":[60],"SOMA$(k,n)$":[62,83],"can":[63,84,113],"constructed":[65],"by":[66],"superposing":[67,166],"$k$":[68,91],"mutually":[69,92],"orthogonal":[70,93],"Latin":[71,94,168],"squares":[72,95],"order":[74,97,171],"$n$":[75],"with":[76,142,150],"pairwise":[77],"disjoint":[78],"symbol-sets,":[79],"so":[81],"seen":[86],"as":[87],"generalization":[89],"$n$.":[98],"In":[99,135],"this":[100],"paper":[101],"we":[102,137],"first":[103],"study":[104],"the":[105,119,129,160],"structure":[106],"SOMAs,":[108],"concentrating":[109],"on":[110,118,173],"how":[111],"SOMAs":[112],"decompose.":[114],"We":[115],"then":[116],"report":[117],"use":[120],"computational":[122],"group":[123],"theory":[124,127],"graph":[126],"discovery":[130],"classification":[132],"SOMAs.":[134],"particular,":[136],"discover":[138,146],"classify":[140],"SOMA$(3,10)$s":[141,163],"certain":[143],"properties,":[144],"two":[147],"SOMA$(4,14)$s":[148],"(SOMAs":[149],"these":[151],"parameters":[152],"were":[153],"previously":[154],"unknown":[155],"to":[156],"exist).":[157],"Some":[158],"newly":[161],"discovered":[162],"come":[164],"from":[165],"square":[169],"10":[172],"SOMA$(2,10)$.":[175]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":3}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
