{"id":"https://openalex.org/W7224901201","doi":"https://doi.org/10.48550/arxiv.2610.39256","title":"On Proximity Gaps for One-Point Algebraic Geometric Codes","display_name":"On Proximity Gaps for One-Point Algebraic Geometric Codes","publication_year":2026,"publication_date":"2026-10-29","ids":{"openalex":"https://openalex.org/W7224901201","doi":"https://doi.org/10.48550/arxiv.2610.39256"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.39256","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.39256","pdf_url":"https://arxiv.org/pdf/2610.39256","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.39256","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151116336","display_name":"Songsong Li","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Li, Songsong","raw_affiliation_strings":["Shanghai Jiao Tong University , Shanghai , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Jiao Tong University , Shanghai , China","institution_ids":["https://openalex.org/I183067930"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080006105","display_name":"Chaoping Xing","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xing, Chaoping","raw_affiliation_strings":["Shanghai Jiao Tong University , Shanghai , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Jiao Tong University , Shanghai , China","institution_ids":["https://openalex.org/I183067930"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5160203275","display_name":"Chen Yuan","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuan, Chen","raw_affiliation_strings":["Shanghai Jiao Tong University , Shanghai , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Jiao Tong University , Shanghai , China","institution_ids":["https://openalex.org/I183067930"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5125220671","display_name":"Ruiqi Zhu","orcid":"https://orcid.org/0009-0009-9917-7351","observed_orcids":["https://orcid.org/0009-0009-9917-7351"]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhu, Ruiqi","raw_affiliation_strings":["Shanghai Jiao Tong University , Shanghai , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Jiao Tong University , Shanghai , China","institution_ids":["https://openalex.org/I183067930"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I183067930"],"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":0.9951000213623047,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9951000213623047,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10237","display_name":"Cryptography and Data Security","score":0.001500000013038516,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.0006000000284984708,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/interactive-oracle-proofs","display_name":"interactive oracle proofs","score":0.9639999866485596},{"id":"https://openalex.org/keywords/soundness","display_name":"soundness","score":0.6980000138282776},{"id":"https://openalex.org/keywords/reed-solomon-codes","display_name":"Reed-Solomon codes","score":0.9629999995231628},{"id":"https://openalex.org/keywords/error-correcting-codes","display_name":"error-correcting codes","score":0.6010000109672546},{"id":"https://openalex.org/keywords/johnson-bound","display_name":"Johnson bound","score":0.6679999828338623},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.45500001311302185}],"concepts":[{"id":"https://openalex.org/C39920170","wikidata":"https://www.wikidata.org/wiki/Q693083","display_name":"Soundness","level":2,"score":0.8805999755859375},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8162000179290771},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.5928999781608582},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5541999936103821},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5284000039100647},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.5210999846458435},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5189999938011169},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4390000104904175},{"id":"https://openalex.org/C55166926","wikidata":"https://www.wikidata.org/wiki/Q2892946","display_name":"Oracle","level":2,"score":0.4212999939918518},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.4072999954223633},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.40619999170303345},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.367000013589859},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.3610000014305115},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.3224000036716461},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.3046000003814697},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.2939999997615814},{"id":"https://openalex.org/C2777831296","wikidata":"https://www.wikidata.org/wiki/Q12518","display_name":"Tower","level":2,"score":0.2897999882698059},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.274399995803833},{"id":"https://openalex.org/C68363185","wikidata":"https://www.wikidata.org/wiki/Q180969","display_name":"Algebraic geometry","level":2,"score":0.2646999955177307},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.2628999948501587},{"id":"https://openalex.org/C9485509","wikidata":"https://www.wikidata.org/wiki/Q1455652","display_name":"Polynomial ring","level":3,"score":0.26100000739097595},{"id":"https://openalex.org/C21714298","wikidata":"https://www.wikidata.org/wiki/Q929536","display_name":"Subring","level":3,"score":0.2597000002861023},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.25769999623298645},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.25609999895095825}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.39256","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.39256","pdf_url":"https://arxiv.org/pdf/2610.39256","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.2610.39256","doi":"https://doi.org/10.48550/arxiv.2610.39256","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.39256","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:2610.39256","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.39256","pdf_url":"https://arxiv.org/pdf/2610.39256","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/W7224901201.pdf","grobid_xml":"https://content.openalex.org/works/W7224901201.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3106277270","https://openalex.org/W7160870119","https://openalex.org/W7164037135","https://openalex.org/W7221101496","https://openalex.org/W2099920789","https://openalex.org/W7168054325","https://openalex.org/W7212157576","https://openalex.org/W3037191715","https://openalex.org/W2067553256","https://openalex.org/W7212021349"],"abstract_inverted_index":{"Proximity":[0],"gaps":[1],"and":[2,43,54,65,132,166,182,198,223],"correlated":[3,59,109,212],"agreement":[4,60,110,213],"underpin":[5],"the":[6,100,150,157,163,190,196,200],"soundness":[7,66,94,116,176,216],"of":[8,31,49,86,97,180],"interactive":[9],"oracle":[10],"proofs.":[11],"We":[12,105],"establish":[13,107],"these":[14],"properties":[15],"for":[16,26,119,214],"general":[17],"one-point":[18],"algebraic":[19],"geometry":[20],"(AG)":[21],"codes,":[22],"extending":[23],"corresponding":[24],"results":[25],"Reed-Solomon":[27],"codes.":[28],"For":[29,46],"codes":[30],"length":[32,88],"$n$":[33],"over":[34,89],"$\\mathbb":[35],"F_q$":[36],"with":[37,61,71,93,115,167,219],"designed":[38,123],"distance":[39,124,205],"$n-m$,":[40],"write":[41],"$\u03c1=m/n$":[42],"fix":[44],"$\u03c1\\in(0,1)$.":[45],"polynomial":[47,139],"curves":[48],"degree":[50],"at":[51,160],"most":[52],"$\\ell$":[53],"any":[55],"$0<\\varepsilon<1$,":[56],"we":[57],"prove":[58],"multiplicative":[62],"slack":[63],"$\\varepsilon$":[64],"$O(\\ell(\\ell+1)/(\\varepsilon":[67],"q))$":[68],"whenever":[69],"$e\\le(1-\\varepsilon)(n-m)/2$,":[70],"no":[72],"additional":[73],"genus":[74],"penalty.":[75],"This":[76],"gives":[77],"asymptotically":[78],"good":[79],"AG":[80],"code":[81],"families":[82],"from":[83],"Garcia--Stichtenoth":[84],"tower":[85,152],"unbounded":[87],"a":[90,137,142,173,185],"fixed":[91],"field":[92,197],"bounds":[95],"independent":[96,179],"$n$,":[98,183],"beyond":[99,155],"RS":[101],"restriction":[102],"$n\\le":[103],"q$.":[104],"also":[106],"mutual":[108],"up":[111],"to":[112,135,207],"radius":[113],"$(1-\\sqrt\u03c1-\u03b7)n$,":[114],"$O(\\ell":[117],"n/(\u03b7^3q))$":[118],"$0<\u03b7<1-\\sqrt\u03c1$.":[120],"Our":[121],"half":[122,156,204],"proof":[125],"combines":[126],"auxiliary":[127],"divisors,":[128],"simultaneous":[129],"Berlekamp-Welch":[130],"interpolation,":[131],"error-incidence":[133],"counting":[134],"obtain":[136],"common":[138],"representation":[140],"without":[141],"Polishchuk-Spielman":[143],"divisibility":[144],"lemma.":[145],"Finally,":[146],"two":[147],"counterexamples":[148],"on":[149],"Garcia-Stichtenoth":[151],"show":[153],"limitations":[154],"minimum":[158],"distance,":[159],"radii":[161],"below":[162],"Johnson":[164,191],"bound":[165,178],"positive":[168,174,220],"slack.":[169],"The":[170,193],"first":[171],"requires":[172],"constant":[175],"lower":[177],"$q$":[181],"precluding":[184],"uniform":[186],"$O(1/q)$":[187],"guarantee":[188],"throughout":[189],"regime.":[192],"second":[194],"fixes":[195],"lets":[199],"relative":[201,224],"gap":[202],"above":[203],"tend":[206],"zero":[208],"while":[209],"ruling":[210],"out":[211],"every":[215],"parameter":[217],"$\u03c3<1$,":[218],"limiting":[221],"rate":[222],"error":[225],"radius.":[226]},"counts_by_year":[],"updated_date":"2026-11-02T09:45:03.842371","created_date":"2026-10-31T00:00:00"}