{"id":"https://openalex.org/W4256426419","doi":"https://doi.org/10.37236/4267","title":"On the Codes Related to the Higman-Sims Graph","display_name":"On the Codes Related to the Higman-Sims Graph","publication_year":2015,"publication_date":"2015-01-27","ids":{"openalex":"https://openalex.org/W4256426419","doi":"https://doi.org/10.37236/4267"},"language":"en","primary_location":{"id":"doi:10.37236/4267","is_oa":true,"landing_page_url":"https://doi.org/10.37236/4267","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v22i1p19/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":true,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v22i1p19/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5086370833","display_name":"Wolfgang Knapp","orcid":null},"institutions":[{"id":"https://openalex.org/I8087733","display_name":"University of T\u00fcbingen","ror":"https://ror.org/03a1kwz48","country_code":"DE","type":"education","lineage":["https://openalex.org/I8087733"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Wolfgang Knapp","raw_affiliation_strings":["University of Tuebingen"],"affiliations":[{"raw_affiliation_string":"University of Tuebingen","institution_ids":["https://openalex.org/I8087733"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5111366165","display_name":"H.-J. Schaeffer","orcid":null},"institutions":[{"id":"https://openalex.org/I8087733","display_name":"University of T\u00fcbingen","ror":"https://ror.org/03a1kwz48","country_code":"DE","type":"education","lineage":["https://openalex.org/I8087733"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Hans-J\u00f6rg Schaeffer","raw_affiliation_strings":["University of Tuebingen"],"affiliations":[{"raw_affiliation_string":"University of Tuebingen","institution_ids":["https://openalex.org/I8087733"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5086370833"],"corresponding_institution_ids":["https://openalex.org/I8087733"],"apc_list":null,"apc_paid":null,"fwci":1.2943,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.87445513,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"22","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9995999932289124,"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":0.9995999932289124,"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/T10796","display_name":"Cooperative Communication and Network Coding","score":0.9840999841690063,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T10849","display_name":"Finite Group Theory Research","score":0.9807000160217285,"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/combinatorics","display_name":"Combinatorics","score":0.7986380457878113},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7973084449768066},{"id":"https://openalex.org/keywords/indecomposable-module","display_name":"Indecomposable module","score":0.4501315951347351},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.4401378929615021},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.419904887676239},{"id":"https://openalex.org/keywords/adjacency-matrix","display_name":"Adjacency matrix","score":0.4189004898071289},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39349132776260376},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.39282965660095215},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.17479237914085388}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7986380457878113},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7973084449768066},{"id":"https://openalex.org/C157480366","wikidata":"https://www.wikidata.org/wiki/Q2499534","display_name":"Indecomposable module","level":2,"score":0.4501315951347351},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.4401378929615021},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.419904887676239},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.4189004898071289},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39349132776260376},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.39282965660095215},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.17479237914085388},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/4267","is_oa":true,"landing_page_url":"https://doi.org/10.37236/4267","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v22i1p19/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":true,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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:10.37236/4267","is_oa":true,"landing_page_url":"https://doi.org/10.37236/4267","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v22i1p19/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":true,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/W4256426419.pdf","grobid_xml":"https://content.openalex.org/works/W4256426419.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W349764376","https://openalex.org/W1491353128","https://openalex.org/W1572931715","https://openalex.org/W1971864279","https://openalex.org/W1972817581","https://openalex.org/W1975652733","https://openalex.org/W2019142182","https://openalex.org/W2032927929","https://openalex.org/W2057973781","https://openalex.org/W2068411303","https://openalex.org/W2073862042","https://openalex.org/W2075906776","https://openalex.org/W2078370438","https://openalex.org/W2083922670","https://openalex.org/W2118368475","https://openalex.org/W2154854282","https://openalex.org/W2917893419","https://openalex.org/W3124200832","https://openalex.org/W4200283578","https://openalex.org/W4233384152","https://openalex.org/W6600618482"],"related_works":["https://openalex.org/W2259692984","https://openalex.org/W4300812862","https://openalex.org/W1664413407","https://openalex.org/W4301018421","https://openalex.org/W2887981140","https://openalex.org/W2896357665","https://openalex.org/W4289388956","https://openalex.org/W2950826653","https://openalex.org/W2374778222","https://openalex.org/W1963961477"],"abstract_inverted_index":{"All":[0],"linear":[1],"codes":[2,119],"of":[3,39,50,98,116,153,196,217,238,269,280],"length":[4],"$100$":[5],"over":[6],"a":[7,177,192,201,236,247],"field":[8],"$F$":[9],"which":[10,160,244],"admit":[11],"the":[12,40,47,51,73,96,108,117,133,147,172,186,205,213,223,229,232,270],"Higman-Sims":[13,52,271],"simple":[14,254],"group":[15,25,180,255,272],"HS":[16,279],"in":[17,87,95,141,158,170,204,234,278,282],"its":[18],"rank":[19],"$3$":[20],"representation":[21,26],"are":[22,102,111,120,128,149],"determined.":[23],"By":[24],"theory":[27],"it":[28,261],"is":[29,56,76,85,176,242,256,262],"proved":[30],"that":[31,174,251,264],"they":[32],"can":[33,273],"all":[34,266],"be":[35,132,274],"understood":[36,275],"as":[37,151,276],"submodules":[38],"permutation":[41],"module":[42,55],"$F\\Omega$":[43,159],"where":[44],"$\\Omega$":[45],"denotes":[46,185,212],"vertex":[48],"set":[49],"graph.":[53],"This":[54],"semisimple":[57],"if":[58,67,183],"$\\mathrm{char}":[59,68],"F\\neq":[60],"2,5$":[61],"and":[62,105,191,210,228],"absolutely":[63],"indecomposable":[64],"otherwise.":[65],"Also":[66],"F":[69],"\\in":[70],"\\{2,":[71],"5\\}$":[72],"submodule":[74],"lattice":[75],"determined":[77,121,135,150],"explicitly.":[78],"The":[79,93,113,166],"binary":[80],"case":[81],"$F":[82],"=":[83,143,220],"\\mathbb{F}_2$":[84],"studied":[86],"detail":[88],"under":[89,181],"coding":[90],"theoretic":[91],"aspects.":[92],"HS-orbits":[94,148,233],"subcodes":[97],"dimension":[99],"$\\leq":[100],"23$":[101],"computed":[103],"explicitly":[104],"so":[106],"also":[107],"weight":[109,114,156,197],"enumerators":[110,115],"obtained.":[112],"dual":[118],"by":[122,136,189],"MacWilliams":[123],"transformation.":[124],"Two":[125],"fundamental":[126],"methods":[127],"used:":[129],"Let":[130],"$v$":[131],"endomorphism":[134],"an":[137],"adjacency":[138],"matrix.":[139],"Then":[140],"$H_{22}":[142],"\\mathrm{Im}":[144],"v":[145],"$":[146],"$v$-images":[152],"certain":[154],"low":[155],"vectors":[157],"carry":[161],"some":[162],"special":[163],"graph":[164],"configurations.":[165],"second":[167],"method":[168],"consists":[169],"using":[171],"fact":[173],"$H_{23}/H_{21}$":[175,227],"Klein":[178],"four":[179],"addition,":[182],"$H_{23}$":[184,235],"code":[187,208],"generated":[188],"$H_{22}$":[190],"\"Higman":[193],"vector\"":[194],"$x(m)$":[195],"50":[198],"associated":[199],"to":[200,246,258],"heptad":[202],"$m$":[203],"shortened":[206],"Golay":[207],"$G_{22}$,":[209],"$H_{21}$":[211],"doubly":[214],"even":[215],"subcode":[216],"$H_{22}\\leq":[218],"H_{78}":[219],"{H_{22}}^\\perp$.":[221],"Using":[222],"mentioned":[224],"observation":[225],"about":[226],"results":[230],"on":[231],"model":[237],"G.":[239,252],"Higman's":[240,253],"geometry":[241],"constructed,":[243],"leads":[245],"direct":[248],"geometric":[249],"proof":[250],"isomorphic":[257],"HS.":[259],"Finally,":[260],"shown":[263],"almost":[265],"maximal":[267],"subgroups":[268],"stabilizers":[277],"codewords":[281],"$H_{23}$.":[283]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}