{"id":"https://openalex.org/W7202149308","doi":"https://doi.org/10.48550/arxiv.2608.06920","title":"Classification of Symmetric Hadamard Matrices Up to Order 32","display_name":"Classification of Symmetric Hadamard Matrices Up to Order 32","publication_year":2026,"publication_date":"2026-08-07","ids":{"openalex":"https://openalex.org/W7202149308","doi":"https://doi.org/10.48550/arxiv.2608.06920"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.06920","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.06920","pdf_url":"https://arxiv.org/pdf/2608.06920","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.06920","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120663493","display_name":"Tuomo Valtonen","orcid":"https://orcid.org/0009-0001-0743-2485","observed_orcids":["https://orcid.org/0009-0001-0743-2485"]},"institutions":[{"id":"https://openalex.org/I9927081","display_name":"Aalto University","ror":"https://ror.org/020hwjq30","country_code":"FI","type":"education","lineage":["https://openalex.org/I9927081"]}],"countries":["FI"],"is_corresponding":true,"raw_author_name":"Valtonen, Tuomo","raw_affiliation_strings":["Department of Information and Communications Engineering Aalto University School of Electrical Engineering P.O. Box 15600 , 00076 Aalto , Finland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Information and Communications Engineering Aalto University School of Electrical Engineering P.O. Box 15600 , 00076 Aalto , Finland","institution_ids":["https://openalex.org/I9927081"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5120663493"],"corresponding_institution_ids":["https://openalex.org/I9927081"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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/T11130","display_name":"Coding theory and cryptography","score":0.5396000146865845,"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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.5396000146865845,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.4011000096797943,"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":0.0333000011742115,"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/weighing-matrices","display_name":"weighing matrices","score":0.8510000109672546}],"concepts":[{"id":"https://openalex.org/C6497535","wikidata":"https://www.wikidata.org/wiki/Q5156556","display_name":"Complex Hadamard matrix","level":4,"score":0.8942999839782715},{"id":"https://openalex.org/C60292330","wikidata":"https://www.wikidata.org/wiki/Q1014065","display_name":"Hadamard transform","level":2,"score":0.8499000072479248},{"id":"https://openalex.org/C31021346","wikidata":"https://www.wikidata.org/wiki/Q1365087","display_name":"Hadamard's inequality","level":4,"score":0.8330000042915344},{"id":"https://openalex.org/C121334634","wikidata":"https://www.wikidata.org/wiki/Q5637579","display_name":"Hadamard's maximal determinant problem","level":5,"score":0.8274000287055969},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.755299985408783},{"id":"https://openalex.org/C30463267","wikidata":"https://www.wikidata.org/wiki/Q1422682","display_name":"Hadamard matrix","level":3,"score":0.7493000030517578},{"id":"https://openalex.org/C68720959","wikidata":"https://www.wikidata.org/wiki/Q3527250","display_name":"Hadamard three-lines theorem","level":4,"score":0.6621999740600586},{"id":"https://openalex.org/C90230998","wikidata":"https://www.wikidata.org/wiki/Q3406746","display_name":"Hadamard product","level":3,"score":0.5950000286102295},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5224999785423279},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5209000110626221},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4296000003814697},{"id":"https://openalex.org/C26693338","wikidata":"https://www.wikidata.org/wiki/Q1567339","display_name":"Hadamard code","level":3,"score":0.4189999997615814},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4034000039100647},{"id":"https://openalex.org/C34459055","wikidata":"https://www.wikidata.org/wiki/Q256151","display_name":"Matrix equivalence","level":4,"score":0.39329999685287476},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.3345000147819519},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33379998803138733},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.3156000077724457},{"id":"https://openalex.org/C35222485","wikidata":"https://www.wikidata.org/wiki/Q729359","display_name":"Hadamard three-circle theorem","level":5,"score":0.31029999256134033},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26989999413490295},{"id":"https://openalex.org/C156103551","wikidata":"https://www.wikidata.org/wiki/Q130998","display_name":"Equivalence relation","level":2,"score":0.2506999969482422}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.06920","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.06920","pdf_url":"https://arxiv.org/pdf/2608.06920","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.06920","doi":"https://doi.org/10.48550/arxiv.2608.06920","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.06920","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.06920","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.06920","pdf_url":"https://arxiv.org/pdf/2608.06920","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7202149308.pdf","grobid_xml":"https://content.openalex.org/works/W7202149308.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W7168135831","https://openalex.org/W7172431312"],"related_works":["https://openalex.org/W7096560042","https://openalex.org/W2131599704","https://openalex.org/W64488262","https://openalex.org/W2075412945","https://openalex.org/W2408229377","https://openalex.org/W2066503751","https://openalex.org/W2001052622","https://openalex.org/W184274398","https://openalex.org/W2795552776","https://openalex.org/W2038351891"],"abstract_inverted_index":{"In":[0,18],"this":[1],"paper,":[2],"symmetric":[3,27,39,70,87],"Hadamard":[4,10,28,40,64,88],"matrices":[5,29,41,79,91],"are":[6,52,92],"classified":[7],"up":[8],"to":[9,68,77],"equivalence":[11],"for":[12],"all":[13],"orders":[14],"at":[15,42],"most":[16],"32.":[17],"particular,":[19],"an":[20],"error":[21],"in":[22],"the":[23,75],"previous":[24],"classification":[25,49],"of":[26,30,37,74],"order":[31,44],"28":[32],"is":[33,45,66,80],"corrected.":[34],"The":[35,48,72],"number":[36],"distinct":[38],"each":[43],"also":[46,81],"enumerated.":[47],"and":[50,83,89],"enumeration":[51],"carried":[53],"out":[54],"using":[55],"a":[56,62,69],"new":[57],"algorithm":[58,76],"that":[59],"determines":[60],"whether":[61],"given":[63],"matrix":[65],"equivalent":[67],"one.":[71],"extension":[73],"weighing":[78,90],"described,":[82],"theoretical":[84],"results":[85],"concerning":[86],"established.":[93]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-11T00:00:00"}
