{"id":"https://openalex.org/W7213904471","doi":"https://doi.org/10.48550/arxiv.2609.21189","title":"A geometric approach to the density of rank-metric codes","display_name":"A geometric approach to the density of rank-metric codes","publication_year":2026,"publication_date":"2026-09-18","ids":{"openalex":"https://openalex.org/W7213904471","doi":"https://doi.org/10.48550/arxiv.2609.21189"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.21189","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21189","pdf_url":"https://arxiv.org/pdf/2609.21189","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/2609.21189","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5071847515","display_name":"Shamil Asgarli","orcid":"https://orcid.org/0000-0003-3165-8297","observed_orcids":["https://orcid.org/0000-0003-3165-8297"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Asgarli, Shamil","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5152331843","display_name":"Lian Duan","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Duan, Lian","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5120811797","display_name":"Nathan Kaplan","orcid":"https://orcid.org/0000-0001-5305-983X","observed_orcids":["https://orcid.org/0000-0001-5305-983X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kaplan, Nathan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5074903759","display_name":"Kuan-Wen Lai","orcid":"https://orcid.org/0000-0002-0071-5781","observed_orcids":["https://orcid.org/0000-0002-0071-5781"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Lai, Kuan-Wen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.5590999722480774,"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"}},"topics":[{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.5590999722480774,"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"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.38109999895095825,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.012500000186264515,"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/rank-metric-codes","display_name":"rank-metric codes","score":0.9950000047683716},{"id":"https://openalex.org/keywords/asymptotic-density","display_name":"asymptotic density","score":0.9380000233650208},{"id":"https://openalex.org/keywords/linear-sections","display_name":"linear sections","score":0.6919999718666077},{"id":"https://openalex.org/keywords/determinantal-varieties","display_name":"determinantal varieties","score":0.7310000061988831},{"id":"https://openalex.org/keywords/chebotarev-density-theorem","display_name":"Chebotarev density theorem","score":0.9179999828338623},{"id":"https://openalex.org/keywords/finite-fields","display_name":"finite fields","score":0.4580000042915344}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7799000144004822},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.6166999936103821},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5687999725341797},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4453999996185303},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.4016000032424927},{"id":"https://openalex.org/C156491182","wikidata":"https://www.wikidata.org/wiki/Q752723","display_name":"Natural density","level":2,"score":0.3691999912261963},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35679998993873596},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.3361000120639801},{"id":"https://openalex.org/C136197465","wikidata":"https://www.wikidata.org/wiki/Q1729295","display_name":"Variety (cybernetics)","level":2,"score":0.32820001244544983},{"id":"https://openalex.org/C177846678","wikidata":"https://www.wikidata.org/wiki/Q1501864","display_name":"Projective test","level":2,"score":0.31380000710487366},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.30959999561309814},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2955999970436096},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.27379998564720154},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.2612000107765198},{"id":"https://openalex.org/C166785042","wikidata":"https://www.wikidata.org/wiki/Q190391","display_name":"Central limit theorem","level":2,"score":0.25949999690055847}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.21189","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21189","pdf_url":"https://arxiv.org/pdf/2609.21189","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.2609.21189","doi":"https://doi.org/10.48550/arxiv.2609.21189","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.21189","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:2609.21189","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21189","pdf_url":"https://arxiv.org/pdf/2609.21189","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/G4071801633","display_name":"Cokernels of Random Matrices and the Geometry of Error-Correcting Codes","funder_award_id":"2154223","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320331164","display_name":"National Science and Technology Council","ror":"https://ror.org/00wnb9798"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213904471.pdf","grobid_xml":"https://content.openalex.org/works/W7213904471.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4389708924"],"related_works":["https://openalex.org/W2890907419","https://openalex.org/W4380875540","https://openalex.org/W3006820729","https://openalex.org/W3088602358","https://openalex.org/W2047445372","https://openalex.org/W4378649699","https://openalex.org/W7154291060","https://openalex.org/W4400488364","https://openalex.org/W4382365175","https://openalex.org/W4412600021"],"abstract_inverted_index":{"We":[0,17],"study":[1],"the":[2,31,36,45,48,63,83,94],"asymptotic":[3],"density":[4,37,85],"of":[5,9,65],"$\\mathbb{F}_q$-point-free":[6],"linear":[7],"sections":[8],"geometrically":[10],"irreducible":[11],"projective":[12],"varieties":[13,69,74,88],"over":[14,89],"finite":[15,90],"fields.":[16],"then":[18],"apply":[19],"these":[20,57],"results":[21],"to":[22,39,67,81,92],"rank-metric":[23],"codes":[24],"via":[25],"determinantal":[26,73],"varieties.":[27],"Our":[28],"approach":[29],"recovers":[30],"known":[32],"cases":[33,49],"in":[34,47],"which":[35],"tends":[38],"$0$":[40],"or":[41],"$1$":[42],"and":[43,70],"determines":[44],"limit":[46],"where":[50],"it":[51],"was":[52],"previously":[53,58],"unknown.":[54],"To":[55],"compute":[56],"unknown":[59],"limits,":[60],"we":[61],"extend":[62],"notion":[64],"quasireflexivity":[66],"higher-dimensional":[68],"show":[71],"that":[72],"satisfy":[75],"this":[76],"property.":[77],"This":[78],"allows":[79],"us":[80],"invoke":[82],"Chebotarev":[84],"theorem":[86],"for":[87],"fields":[91],"obtain":[93],"desired":[95],"estimate.":[96]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-22T00:00:00"}