{"id":"https://openalex.org/W7212126167","doi":"https://doi.org/10.48550/arxiv.2609.10274","title":"Revisiting the Weight Spectrum of the Affine Grassmann Code $C^{\\mathbb{A}}(2,m)$","display_name":"Revisiting the Weight Spectrum of the Affine Grassmann Code $C^{\\mathbb{A}}(2,m)$","publication_year":2026,"publication_date":"2026-09-09","ids":{"openalex":"https://openalex.org/W7212126167","doi":"https://doi.org/10.48550/arxiv.2609.10274"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.10274","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10274","pdf_url":"https://arxiv.org/pdf/2609.10274","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/2609.10274","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5115464408","display_name":"Rohit Yadav","orcid":"https://orcid.org/0000-0002-6200-0108","observed_orcids":["https://orcid.org/0000-0002-6200-0108"]},"institutions":[{"id":"https://openalex.org/I4210127441","display_name":"Indian Institute of Technology Jammu","ror":"https://ror.org/02f0vsw63","country_code":"IN","type":"education","lineage":["https://openalex.org/I4210127441"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Yadav, Rohit","raw_affiliation_strings":["Department of Mathematics , Indian Institute of Technology , Jammu , Jammu- 181221 , India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Indian Institute of Technology , Jammu , Jammu- 181221 , India","institution_ids":["https://openalex.org/I4210127441"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5115464408"],"corresponding_institution_ids":["https://openalex.org/I4210127441"],"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":0.9994999766349792,"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":0.00019999999494757503,"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/weight-spectrum","display_name":"weight spectrum","score":0.7850000262260437},{"id":"https://openalex.org/keywords/hamming-weight","display_name":"Hamming weight","score":0.9340000152587891},{"id":"https://openalex.org/keywords/linear-codes-over-finite-fields","display_name":"linear codes over finite fields","score":0.8009999990463257}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7993999719619751},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.7389000058174133},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.722000002861023},{"id":"https://openalex.org/C207221997","wikidata":"https://www.wikidata.org/wiki/Q938614","display_name":"Affine combination","level":3,"score":0.6647999882698059},{"id":"https://openalex.org/C112128483","wikidata":"https://www.wikidata.org/wiki/Q382504","display_name":"Affine hull","level":4,"score":0.5877000093460083},{"id":"https://openalex.org/C63361517","wikidata":"https://www.wikidata.org/wiki/Q5645805","display_name":"Hamming weight","level":5,"score":0.5594000220298767},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.46860000491142273},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4456000030040741},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4429999887943268},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.43799999356269836},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.4275999963283539},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.4108000099658966},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.40709999203681946},{"id":"https://openalex.org/C173110770","wikidata":"https://www.wikidata.org/wiki/Q382698","display_name":"Affine space","level":3,"score":0.4049000144004822},{"id":"https://openalex.org/C2400350","wikidata":"https://www.wikidata.org/wiki/Q1752667","display_name":"Linear code","level":4,"score":0.390500009059906},{"id":"https://openalex.org/C2096984","wikidata":"https://www.wikidata.org/wiki/Q245464","display_name":"Cyclic code","level":5,"score":0.37779998779296875},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3675000071525574},{"id":"https://openalex.org/C154610627","wikidata":"https://www.wikidata.org/wiki/Q382491","display_name":"Affine plane (incidence geometry)","level":3,"score":0.3474000096321106},{"id":"https://openalex.org/C174072685","wikidata":"https://www.wikidata.org/wiki/Q1667469","display_name":"Prime power","level":3,"score":0.34049999713897705},{"id":"https://openalex.org/C95565831","wikidata":"https://www.wikidata.org/wiki/Q382698","display_name":"Affine coordinate system","level":4,"score":0.3264999985694885},{"id":"https://openalex.org/C185579658","wikidata":"https://www.wikidata.org/wiki/Q3554813","display_name":"Affine variety","level":3,"score":0.3140000104904175},{"id":"https://openalex.org/C62623296","wikidata":"https://www.wikidata.org/wiki/Q686291","display_name":"Affine group","level":3,"score":0.3052999973297119},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29829999804496765},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.296099990606308},{"id":"https://openalex.org/C172331165","wikidata":"https://www.wikidata.org/wiki/Q524607","display_name":"General linear group","level":3,"score":0.2930000126361847},{"id":"https://openalex.org/C198082693","wikidata":"https://www.wikidata.org/wiki/Q1551631","display_name":"Gr\u00f6bner basis","level":3,"score":0.29109999537467957},{"id":"https://openalex.org/C20050597","wikidata":"https://www.wikidata.org/wiki/Q261527","display_name":"Linear form","level":2,"score":0.27790001034736633},{"id":"https://openalex.org/C6128414","wikidata":"https://www.wikidata.org/wiki/Q3892887","display_name":"Affine representation","level":5,"score":0.26969999074935913},{"id":"https://openalex.org/C23684344","wikidata":"https://www.wikidata.org/wiki/Q382520","display_name":"Affine geometry","level":4,"score":0.26809999346733093},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.26109999418258667},{"id":"https://openalex.org/C2778945305","wikidata":"https://www.wikidata.org/wiki/Q6865502","display_name":"Minimum weight","level":2,"score":0.25769999623298645}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.10274","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10274","pdf_url":"https://arxiv.org/pdf/2609.10274","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.2609.10274","doi":"https://doi.org/10.48550/arxiv.2609.10274","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.10274","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:2609.10274","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10274","pdf_url":"https://arxiv.org/pdf/2609.10274","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212126167.pdf","grobid_xml":"https://content.openalex.org/works/W7212126167.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7212017002","https://openalex.org/W2899403201","https://openalex.org/W2964056253","https://openalex.org/W6884296901","https://openalex.org/W1787244719","https://openalex.org/W628735373","https://openalex.org/W2794724414","https://openalex.org/W1800035742","https://openalex.org/W3005217436","https://openalex.org/W2038442868"],"abstract_inverted_index":{"Affine":[0],"Grassmann":[1,31],"codes,":[2],"introduced":[3],"by":[4,15,36],"Beelen,":[5],"Ghorpade":[6],"and":[7,38,56,64,80,96],"H\u00f8holdt,":[8],"are":[9],"linear":[10,17],"codes":[11],"over":[12],"$\\mathbb{F}_q$":[13],"obtained":[14],"evaluating":[16],"combinations":[18],"of":[19,21,28,47],"minors":[20],"a":[22,41,76,87],"generic":[23],"matrix.":[24,51],"The":[25],"weight":[26,83],"spectrum":[27],"the":[29,45],"affine":[30,94],"code":[32],"$C^{\\mathbb{A}}(2,m)$":[33],"was":[34],"determined":[35],"Pi\u00f1ero":[37],"Singh":[39],"via":[40],"case":[42],"analysis":[43],"on":[44],"rank":[46],"an":[48,54],"associated":[49],"alternating":[50],"We":[52],"give":[53],"independent":[55],"more":[57],"streamlined":[58],"derivation,":[59],"valid":[60],"for":[61],"all":[62],"$m\\ge4$":[63],"every":[65],"prime":[66],"power":[67],"$q$,":[68],"in":[69,75],"which":[70],"each":[71],"codeword":[72],"is":[73,84],"written":[74],"compact":[77],"matrix":[78],"form":[79],"its":[81],"Hamming":[82],"expressed":[85],"through":[86],"single":[88],"closed":[89],"formula":[90],"involving":[91],"two":[92],"explicit":[93],"subspaces":[95],"their":[97],"intersection.":[98]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-11T00:00:00"}