{"id":"https://openalex.org/W7160939758","doi":"https://doi.org/10.48550/arxiv.2605.08973","title":"Tight Lower Bounds on The Single-Error Detection Threshold for Analog Error-Correcting Codes","display_name":"Tight Lower Bounds on The Single-Error Detection Threshold for Analog Error-Correcting Codes","publication_year":2026,"publication_date":"2026-05-09","ids":{"openalex":"https://openalex.org/W7160939758","doi":"https://doi.org/10.48550/arxiv.2605.08973"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.08973","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.08973","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.08973","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5135936881","display_name":"Zhengyi Jiang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jiang, Zhengyi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5135928569","display_name":"Wenhao Liu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Liu, Wenhao","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5135917226","display_name":"Zhongyi Huang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Huang, Zhongyi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5001405009","display_name":"Bo Bai","orcid":"https://orcid.org/0000-0003-4796-8249"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bai, Bo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5135954816","display_name":"Gong Zhang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhang, Gong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5039863234","display_name":"Hanxu Hou","orcid":"https://orcid.org/0000-0001-7328-9341"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hou, Hanxu","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":false,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.1436000019311905,"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"}},"topics":[{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.1436000019311905,"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"}},{"id":"https://openalex.org/T10772","display_name":"Distributed systems and fault tolerance","score":0.13130000233650208,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11005","display_name":"Radiation Effects in Electronics","score":0.11720000207424164,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5978000164031982},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.557200014591217},{"id":"https://openalex.org/keywords/coding-theory","display_name":"Coding theory","score":0.45019999146461487},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.4153999984264374},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.40560001134872437},{"id":"https://openalex.org/keywords/code","display_name":"Code (set theory)","score":0.37700000405311584},{"id":"https://openalex.org/keywords/multiplication","display_name":"Multiplication (music)","score":0.3547999858856201}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6771000027656555},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5978000164031982},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.557200014591217},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.524399995803833},{"id":"https://openalex.org/C113709454","wikidata":"https://www.wikidata.org/wiki/Q602136","display_name":"Coding theory","level":2,"score":0.45019999146461487},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4442000091075897},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.4153999984264374},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.40560001134872437},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.37700000405311584},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.3547999858856201},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.33899998664855957},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.3199999928474426},{"id":"https://openalex.org/C179518139","wikidata":"https://www.wikidata.org/wiki/Q5140297","display_name":"Coding (social sciences)","level":2,"score":0.2888000011444092},{"id":"https://openalex.org/C166366890","wikidata":"https://www.wikidata.org/wiki/Q602136","display_name":"Channel code","level":3,"score":0.27390000224113464},{"id":"https://openalex.org/C137270730","wikidata":"https://www.wikidata.org/wiki/Q120811","display_name":"Detection theory","level":3,"score":0.27079999446868896},{"id":"https://openalex.org/C63361517","wikidata":"https://www.wikidata.org/wiki/Q5645805","display_name":"Hamming weight","level":5,"score":0.26919999718666077},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.2689000070095062},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.26809999346733093},{"id":"https://openalex.org/C2776502983","wikidata":"https://www.wikidata.org/wiki/Q690182","display_name":"Contrast (vision)","level":2,"score":0.25440001487731934}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.08973","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.08973","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.08973","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.08973","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"},"sustainable_development_goals":[{"score":0.6292731761932373,"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Analog":[0,56,183,258],"error-correcting":[1],"codes":[2],"(Analog":[3],"ECCs)":[4],"for":[5,23,83,100,182,235,254],"approximate":[6],"vector-matrix":[7],"multiplication":[8],"have":[9,38],"been":[10,39],"extensively":[11],"studied":[12],"as":[13,133],"means":[14],"to":[15,63,173,188,195,205,215,224],"achieve":[16],"fault-tolerant":[17],"in-memory":[18],"computation.":[19],"The":[20],"theoretical":[21],"foundations":[22],"such":[24],"coding":[25],"schemes,":[26],"particularly":[27],"the":[28,35,51,77,105,108,119,128,152,155,174,178,227,236,245,248],"characterization":[29],"of":[30,53,79,115,123,142,162,252],"their":[31],"correction":[32],"capabilities":[33],"via":[34],"height":[36],"profile,":[37],"well":[40],"established":[41],"in":[42,71,136,150,230,247,257],"recent":[43],"literature.":[44],"In":[45,197],"this":[46,64,198],"paper,":[47,199],"we":[48,200,219],"focus":[49],"on":[50,177],"case":[52,65,122,237],"single-error":[54,179,255],"detection":[55,180,256],"ECCs.":[57,259],"Among":[58],"several":[59],"open":[60],"problems":[61,170],"related":[62,214],"proposed":[66],"by":[67],"Ron":[68],"M.":[69],"Roth":[70],"[1],":[72],"Problem":[73,124,134,185,192,206,231],"1":[74,125,186,232],"asks:":[75],"\"Identify":[76],"values":[78,114,161],"$k$":[80],"and":[81,110,157,191,208],"$n$":[82,144],"which":[84,151],"every":[85,139],"linear":[86],"$[n,":[87],"k]$":[88],"code":[89],"$\\mathcal{C}$":[90],"over":[91],"$\\mathbb{R}$":[92],"satisfies:":[93],"$$\\mathsf{h}_1(\\mathcal{C}):=\\max_{\\boldsymbol{c}\\in":[94],"\\mathcal{C}\\setminus{\\{\\boldsymbol{0}\\}}}\\mathsf{h}_1(\\boldsymbol{c})\\geq":[95],"\\Big\\lceil":[96],"\\frac{k}{n-k}":[97],"\\Big\\rceil.\\text{\"}$$":[98],"Here,":[99],"any":[101],"$\\boldsymbol{x}\\in\\mathbb{R}^n$,":[102],"$\\mathsf{h}_1(\\boldsymbol{x})$":[103],"represents":[104],"ratio":[106,153],"between":[107,154],"largest":[109,112,156,159],"second":[111,158],"absolute":[113,160],"$\\boldsymbol{x}$'s":[116],"entries.":[117],"As":[118],"simplest":[120],"special":[121],"(with":[126],"$n-k=2$),":[127],"following":[129],"problem":[130],"was":[131],"posed":[132],"2":[135,193,207],"[1]:":[137],"\"Must":[138],"$(n-2)$-dimensional":[140],"subspace":[141],"$\\mathbb{R}^n$,":[143],"even,":[145],"contain":[146],"a":[147,209],"nonzero":[148],"vector":[149],"its":[163],"entries":[164],"is":[165,233],"at":[166],"least":[167],"$(n/2)-1$?\"":[168],"These":[169],"directly":[171],"pertain":[172],"lower":[175,228,249],"bounds":[176],"threshold":[181],"ECCs:":[184],"corresponds":[187,194],"arbitrary":[189],"$n-k$":[190,239],"$n-k=2$.":[196],"provide":[201],"an":[202],"affirmative":[203],"answer":[204],"rigorous":[210],"proof":[211],"using":[212],"theories":[213],"convex":[216],"optimization.":[217],"Furthermore,":[218],"extend":[220],"our":[221],"analytical":[222],"method":[223],"show":[225],"that":[226],"bound":[229,250],"tight":[234],"where":[238],"divides":[240],"$k$.":[241],"Our":[242],"results":[243],"fill":[244],"gap":[246],"theory":[251],"thresholds":[253]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-05-13T00:00:00"}