{"id":"https://openalex.org/W7215746358","doi":"https://doi.org/10.48550/arxiv.2610.00142","title":"The kernel-block rank profiles of the $\\mathbb{Z}_2\\mathbb{Z}_4$-linear and the $\\mathbb{Z}_{2^s}$-linear Hadamard codes, and a complete classification of the $\\mathbb{Z}_2\\mathbb{Z}_4\\mathbb{Z}_8$-linear Hadamard codes","display_name":"The kernel-block rank profiles of the $\\mathbb{Z}_2\\mathbb{Z}_4$-linear and the $\\mathbb{Z}_{2^s}$-linear Hadamard codes, and a complete classification of the $\\mathbb{Z}_2\\mathbb{Z}_4\\mathbb{Z}_8$-linear Hadamard codes","publication_year":2026,"publication_date":"2026-09-10","ids":{"openalex":"https://openalex.org/W7215746358","doi":"https://doi.org/10.48550/arxiv.2610.00142"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.00142","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00142","pdf_url":"https://arxiv.org/pdf/2610.00142","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/2610.00142","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5011940850","display_name":"Dipak K. Bhunia","orcid":"https://orcid.org/0000-0003-4852-8739","observed_orcids":["https://orcid.org/0000-0003-4852-8739"]},"institutions":[{"id":"https://openalex.org/I9617848","display_name":"Universitat Polit\u00e8cnica de Catalunya","ror":"https://ror.org/03mb6wj31","country_code":"ES","type":"education","lineage":["https://openalex.org/I9617848"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Bhunia, Dipak K.","raw_affiliation_strings":["Departament de Matem\u00e0tiques , Universitat Polit\u00e8cnica de Catalunya , Barcelona , Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departament de Matem\u00e0tiques , Universitat Polit\u00e8cnica de Catalunya , Barcelona , Spain","institution_ids":["https://openalex.org/I9617848"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5011940850"],"corresponding_institution_ids":["https://openalex.org/I9617848"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":3,"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":0.9994999766349792,"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":0.9994999766349792,"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":9.999999747378752e-05,"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":0.0,"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/nonlinear-codes","display_name":"nonlinear codes","score":0.47600001096725464},{"id":"https://openalex.org/keywords/code-classification","display_name":"code classification","score":0.6959999799728394},{"id":"https://openalex.org/keywords/puncturing","display_name":"puncturing","score":0.6110000014305115},{"id":"https://openalex.org/keywords/kernel-dimension","display_name":"kernel dimension","score":0.5640000104904175},{"id":"https://openalex.org/keywords/binary-codes","display_name":"binary codes","score":0.492000013589859},{"id":"https://openalex.org/keywords/z4-linear-codes","display_name":"Z4-linear codes","score":0.7390000224113464}],"concepts":[{"id":"https://openalex.org/C60292330","wikidata":"https://www.wikidata.org/wiki/Q1014065","display_name":"Hadamard transform","level":2,"score":0.8125},{"id":"https://openalex.org/C26693338","wikidata":"https://www.wikidata.org/wiki/Q1567339","display_name":"Hadamard code","level":3,"score":0.7588000297546387},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7494000196456909},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5615000128746033},{"id":"https://openalex.org/C2400350","wikidata":"https://www.wikidata.org/wiki/Q1752667","display_name":"Linear code","level":4,"score":0.4731999933719635},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.4535999894142151},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.447299987077713},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43560001254081726},{"id":"https://openalex.org/C68720959","wikidata":"https://www.wikidata.org/wiki/Q3527250","display_name":"Hadamard three-lines theorem","level":4,"score":0.4092999994754791},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.4043000042438507},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.3799000084400177},{"id":"https://openalex.org/C63435697","wikidata":"https://www.wikidata.org/wiki/Q864135","display_name":"Binary code","level":3,"score":0.36169999837875366},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.33239999413490295},{"id":"https://openalex.org/C31021346","wikidata":"https://www.wikidata.org/wiki/Q1365087","display_name":"Hadamard's inequality","level":4,"score":0.32260000705718994},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.3154999911785126},{"id":"https://openalex.org/C73666251","wikidata":"https://www.wikidata.org/wiki/Q5310189","display_name":"Dual code","level":5,"score":0.3046000003814697},{"id":"https://openalex.org/C30463267","wikidata":"https://www.wikidata.org/wiki/Q1422682","display_name":"Hadamard matrix","level":3,"score":0.28119999170303345},{"id":"https://openalex.org/C6497535","wikidata":"https://www.wikidata.org/wiki/Q5156556","display_name":"Complex Hadamard matrix","level":4,"score":0.27970001101493835}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.00142","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00142","pdf_url":"https://arxiv.org/pdf/2610.00142","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.2610.00142","doi":"https://doi.org/10.48550/arxiv.2610.00142","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.00142","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:2610.00142","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00142","pdf_url":"https://arxiv.org/pdf/2610.00142","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":[{"id":"https://openalex.org/G1105844351","display_name":null,"funder_award_id":"10.13039.501100011033","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"},{"id":"https://openalex.org/G491712853","display_name":null,"funder_award_id":"/ AEI10.13039/501100011033","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"},{"id":"https://openalex.org/G5635532810","display_name":null,"funder_award_id":"13039","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"},{"id":"https://openalex.org/G593054264","display_name":null,"funder_award_id":"MICIU/AEI/ 10.13039/501,100,011,033 /","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"},{"id":"https://openalex.org/G7395345766","display_name":null,"funder_award_id":"MICIU/AEI/10.13039","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"},{"id":"https://openalex.org/G7805626310","display_name":null,"funder_award_id":"10.13039","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"},{"id":"https://openalex.org/G7863371703","display_name":null,"funder_award_id":".13039/501100011033","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"},{"id":"https://openalex.org/G8497050386","display_name":null,"funder_award_id":"501100011033","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"},{"id":"https://openalex.org/G8667377315","display_name":null,"funder_award_id":"MICIU/AEI/10","funder_id":"https://openalex.org/F4320315062","funder_display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades"}],"funders":[{"id":"https://openalex.org/F4320315062","display_name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","ror":null},{"id":"https://openalex.org/F4320335598","display_name":"Agencia Estatal de Investigaci\u00f3n","ror":"https://ror.org/003x0zc53"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7215746358.pdf","grobid_xml":"https://content.openalex.org/works/W7215746358.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W7212073272","https://openalex.org/W7212159523"],"related_works":["https://openalex.org/W7212159523","https://openalex.org/W7212073272","https://openalex.org/W7100087132","https://openalex.org/W2957308865","https://openalex.org/W2081939650","https://openalex.org/W2990636281","https://openalex.org/W2903702749","https://openalex.org/W2892003982","https://openalex.org/W4294881352","https://openalex.org/W4401692779"],"abstract_inverted_index":{"The":[0,116,230],"kernel":[1,25,138,166,196,240],"of":[2,41,44,56,70,214,232],"a":[3,67,150,248],"binary":[4,12],"code":[5,104,111,132,162,172,205,213],"containing":[6],"the":[7,11,20,28,34,39,42,53,71,80,131,142,161,169,176,191,195,215,220,238,243],"zero":[8],"word":[9,26],"partitions":[10],"coordinates":[13,17],"into":[14],"blocks,":[15,139],"two":[16],"lying":[18],"in":[19,31,145,237],"same":[21,29,216],"block":[22,167],"when":[23],"every":[24,100,107,201,207],"takes":[27],"value":[30],"both,":[32],"and":[33,65,82,92,105,133,158,183,193,210,225,242],"\\emph{kernel-block":[35],"rank":[36,123,179,245],"profile}":[37],"is":[38,62,97,119,156,168,184,190],"multiset":[40],"dimensions":[43],"its":[45,137],"linear":[46],"span":[47],"punctured":[48,163],"on":[49,136,164],"those":[50,88],"blocks.":[51],"For":[52],"family":[54,223,235],"$H^{t_1,t_2,t_3}$":[55],"$\\mathbb{Z}_2\\mathbb{Z}_4\\mathbb{Z}_8$-linear":[57,203],"Hadamard":[58,84,103,110,171,204,212],"codes,":[59],"this":[60,74,233],"invariant":[61],"known":[63],"explicitly":[64],"gives":[66],"complete":[68],"classification":[69],"family.":[72],"In":[73],"paper,":[75],"we":[76,93],"compute":[77],"it":[78,96],"for":[79,99,106,219],"$\\mathbb{Z}_2\\mathbb{Z}_4$-linear":[81,102,209],"$\\mathbb{Z}_{2^s}$-linear":[83,109,211],"families":[85],"with":[86,114,228],"which":[87,140],"codes":[89],"are":[90],"compared,":[91],"prove":[94,149],"that":[95,129],"constant":[98,177],"nonlinear":[101,108,157,202],"$\\bar":[112,154,173,226],"H^{a_1,\\dots,a_s}$":[113,155],"$s\\geq2$.":[115],"second":[117],"statement":[118],"obtained":[120],"without":[121],"any":[122],"formula,":[124],"by":[125],"exhibiting":[126],"coordinate":[127],"permutations":[128],"preserve":[130],"act":[134],"transitively":[135],"makes":[141],"argument":[143],"uniform":[144],"$s$.":[146],"We":[147],"also":[148],"descent":[151],"theorem:":[152],"if":[153],"$a_1\\geq2$,":[159],"then":[160],"one":[165],"$\\mathbb{Z}_{2^{s-1}}$-linear":[170],"H^{a_1,\\dots,a_{s-1}}$.":[174],"Consequently,":[175],"local":[178],"equals":[180],"$rank(\\bar":[181],"H^{a_1,\\dots,a_{s-1}})$":[182],"at":[185],"least":[186],"$t-\u03ba+2$,":[187],"where":[188],"$2^t$":[189],"length":[192],"$\u03ba$":[194],"dimension.":[197],"These":[198],"results":[199],"separate":[200],"from":[206],"$\\mathbb{Z}_4$-linear,":[208],"length,":[217],"except":[218],"single":[221],"infinite":[222,234],"$H^{1,1,t-4}$":[224],"H^{2,0,t-5}$":[227],"$t\\geq5$.":[229],"members":[231],"agree":[236],"rank,":[239],"dimension":[241],"kernel-block":[244],"profile,":[246],"but":[247],"two-block":[249],"refinement":[250],"separates":[251],"them.":[252]},"counts_by_year":[{"year":2026,"cited_by_count":3}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-03T00:00:00"}