{"id":"https://openalex.org/W7212017002","doi":"https://doi.org/10.48550/arxiv.2609.08784","title":"The Weight Spectrum of the Affine Grassmann Code $C^{\\mathbb A}(3,6)$","display_name":"The Weight Spectrum of the Affine Grassmann Code $C^{\\mathbb A}(3,6)$","publication_year":2026,"publication_date":"2026-09-08","ids":{"openalex":"https://openalex.org/W7212017002","doi":"https://doi.org/10.48550/arxiv.2609.08784"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.08784","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.08784","pdf_url":"https://arxiv.org/pdf/2609.08784","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/2609.08784","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029898403","display_name":"Prasant Singh","orcid":"https://orcid.org/0000-0003-4400-4858","observed_orcids":["https://orcid.org/0000-0003-4400-4858"]},"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":false,"raw_author_name":"Singh, Prasant","raw_affiliation_strings":["Department of Mathematics , Indian Institute of Technology , Jammu , India -181221 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Indian Institute of Technology , Jammu , India -181221 ","institution_ids":["https://openalex.org/I4210127441"]}]},{"author_position":"last","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":false,"raw_author_name":"Yadav, Rohit","raw_affiliation_strings":["Department of Mathematics , Indian Institute of Technology , Jammu , India -181221 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Indian Institute of Technology , Jammu , India -181221 ","institution_ids":["https://openalex.org/I4210127441"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"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":0.9897000193595886,"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":0.9897000193595886,"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":0.007499999832361937,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.00039999998989515007,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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":0.7519999742507935},{"id":"https://openalex.org/keywords/hamming-weight-distribution","display_name":"Hamming weight distribution","score":0.7870000004768372},{"id":"https://openalex.org/keywords/matrix-minors","display_name":"matrix minors","score":0.6980000138282776},{"id":"https://openalex.org/keywords/grassmannians","display_name":"Grassmannians","score":0.7319999933242798},{"id":"https://openalex.org/keywords/linear-codes","display_name":"linear codes","score":0.5429999828338623}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7803000211715698},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.7095999717712402},{"id":"https://openalex.org/C112128483","wikidata":"https://www.wikidata.org/wiki/Q382504","display_name":"Affine hull","level":4,"score":0.6279000043869019},{"id":"https://openalex.org/C207221997","wikidata":"https://www.wikidata.org/wiki/Q938614","display_name":"Affine combination","level":3,"score":0.6017000079154968},{"id":"https://openalex.org/C63361517","wikidata":"https://www.wikidata.org/wiki/Q5645805","display_name":"Hamming weight","level":5,"score":0.5934000015258789},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.4918999969959259},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.48750001192092896},{"id":"https://openalex.org/C162929932","wikidata":"https://www.wikidata.org/wiki/Q129638","display_name":"Grassmannian","level":2,"score":0.46399998664855957},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.43630000948905945},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.42669999599456787},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42239999771118164},{"id":"https://openalex.org/C154610627","wikidata":"https://www.wikidata.org/wiki/Q382491","display_name":"Affine plane (incidence geometry)","level":3,"score":0.39489999413490295},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.3889000117778778},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.3865000009536743},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.35910001397132874},{"id":"https://openalex.org/C2778945305","wikidata":"https://www.wikidata.org/wiki/Q6865502","display_name":"Minimum weight","level":2,"score":0.33070001006126404},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.3260999917984009},{"id":"https://openalex.org/C62623296","wikidata":"https://www.wikidata.org/wiki/Q686291","display_name":"Affine group","level":3,"score":0.32030001282691956},{"id":"https://openalex.org/C103275481","wikidata":"https://www.wikidata.org/wiki/Q6787889","display_name":"Matrix representation","level":3,"score":0.3181999921798706},{"id":"https://openalex.org/C2779760435","wikidata":"https://www.wikidata.org/wiki/Q5396169","display_name":"Minor (academic)","level":2,"score":0.311599999666214},{"id":"https://openalex.org/C2096984","wikidata":"https://www.wikidata.org/wiki/Q245464","display_name":"Cyclic code","level":5,"score":0.29899999499320984},{"id":"https://openalex.org/C173110770","wikidata":"https://www.wikidata.org/wiki/Q382698","display_name":"Affine space","level":3,"score":0.29589998722076416},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2921000123023987},{"id":"https://openalex.org/C95565831","wikidata":"https://www.wikidata.org/wiki/Q382698","display_name":"Affine coordinate system","level":4,"score":0.2549999952316284},{"id":"https://openalex.org/C2400350","wikidata":"https://www.wikidata.org/wiki/Q1752667","display_name":"Linear code","level":4,"score":0.2508000135421753}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.08784","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.08784","pdf_url":"https://arxiv.org/pdf/2609.08784","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.2609.08784","doi":"https://doi.org/10.48550/arxiv.2609.08784","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.08784","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:2609.08784","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.08784","pdf_url":"https://arxiv.org/pdf/2609.08784","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/W7212017002.pdf","grobid_xml":"https://content.openalex.org/works/W7212017002.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7212126167","https://openalex.org/W2899403201","https://openalex.org/W2964056253","https://openalex.org/W628735373","https://openalex.org/W2046856711","https://openalex.org/W2038442868","https://openalex.org/W7211913667","https://openalex.org/W3005217436","https://openalex.org/W7025543135","https://openalex.org/W2031282377"],"abstract_inverted_index":{"In":[0],"this":[1,59],"article,":[2],"we":[3,61,85],"consider":[4],"the":[5,13,20,25,47,77,87,92],"affine":[6,14,93],"Grassmann":[7,94],"code":[8,95],"$C^{\\mathbb":[9,70,96],"A}(3,6)$,":[10],"obtained":[11],"from":[12],"open":[15],"cell":[16],"${\\mathbb":[17],"A}^9$":[18],"of":[19,27,32,34,37,50,67,69,79,91],"Grassmannian":[21],"$G_{3,6}$.":[22],"We":[23],"exploit":[24],"representation":[26],"codewords":[28,68,80],"as":[29],"linear":[30],"combinations":[31],"minors":[33],"all":[35,63],"sizes":[36],"a":[38,51,55],"generic":[39],"$3\\times3$":[40],"matrix":[41],"and":[42],"classify":[43],"them":[44],"according":[45],"to":[46],"largest":[48],"size":[49],"minor":[52],"occurring":[53],"with":[54],"nonzero":[56],"coefficient.":[57],"Using":[58],"classification,":[60],"determine":[62],"possible":[64],"Hamming":[65],"weights":[66],"A}(3,6)$":[71],"and,":[72],"for":[73],"each":[74],"weight,":[75],"compute":[76],"number":[78],"attaining":[81],"that":[82],"weight.":[83],"Consequently,":[84],"obtain":[86],"complete":[88],"weight":[89],"spectrum":[90],"A}(3,6)$.":[97]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-10T00:00:00"}
