{"id":"https://openalex.org/W7161704532","doi":"https://doi.org/10.48550/arxiv.2605.18323","title":"Existence and Counting Bounds for High-Memory Spatially-Coupled Codes via the Combinatorial Nullstellensatz","display_name":"Existence and Counting Bounds for High-Memory Spatially-Coupled Codes via the Combinatorial Nullstellensatz","publication_year":2026,"publication_date":"2026-05-18","ids":{"openalex":"https://openalex.org/W7161704532","doi":"https://doi.org/10.48550/arxiv.2605.18323"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.18323","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.18323","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,"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.2605.18323","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5136476455","display_name":"Lei Huang","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Huang, Lei","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5136476455"],"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/T11321","display_name":"Error Correcting Code Techniques","score":0.961899995803833,"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"}},"topics":[{"id":"https://openalex.org/T11321","display_name":"Error Correcting Code Techniques","score":0.961899995803833,"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/T11181","display_name":"Advanced Data Storage Technologies","score":0.007300000172108412,"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/T10796","display_name":"Cooperative Communication and Network Coding","score":0.00430000014603138,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5902000069618225},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.5034000277519226},{"id":"https://openalex.org/keywords/constructive","display_name":"Constructive","score":0.48890000581741333},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.47049999237060547},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.4438000023365021},{"id":"https://openalex.org/keywords/base","display_name":"Base (topology)","score":0.40380001068115234},{"id":"https://openalex.org/keywords/focus","display_name":"Focus (optics)","score":0.3944000005722046},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.3824999928474426},{"id":"https://openalex.org/keywords/code","display_name":"Code (set theory)","score":0.3758000135421753}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7405999898910522},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5902000069618225},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.532800018787384},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.5034000277519226},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.48890000581741333},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4830000102519989},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.47049999237060547},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.4438000023365021},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.40380001068115234},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.3944000005722046},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.3824999928474426},{"id":"https://openalex.org/C2776760102","wikidata":"https://www.wikidata.org/wiki/Q5139990","display_name":"Code (set theory)","level":3,"score":0.3758000135421753},{"id":"https://openalex.org/C122451846","wikidata":"https://www.wikidata.org/wiki/Q5150824","display_name":"Combinatorial design","level":2,"score":0.35530000925064087},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.35429999232292175},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.35120001435279846},{"id":"https://openalex.org/C2777434295","wikidata":"https://www.wikidata.org/wiki/Q959831","display_name":"Girth (graph theory)","level":2,"score":0.35089999437332153},{"id":"https://openalex.org/C57273362","wikidata":"https://www.wikidata.org/wiki/Q576722","display_name":"Decoding methods","level":2,"score":0.3492000102996826},{"id":"https://openalex.org/C125411270","wikidata":"https://www.wikidata.org/wiki/Q18653","display_name":"Encoding (memory)","level":2,"score":0.3334999978542328},{"id":"https://openalex.org/C166530166","wikidata":"https://www.wikidata.org/wiki/Q1970635","display_name":"Hamming bound","level":5,"score":0.3330000042915344},{"id":"https://openalex.org/C2776502983","wikidata":"https://www.wikidata.org/wiki/Q690182","display_name":"Contrast (vision)","level":2,"score":0.3246999979019165},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.31450000405311584},{"id":"https://openalex.org/C202854965","wikidata":"https://www.wikidata.org/wiki/Q3044470","display_name":"Constructive proof","level":2,"score":0.30959999561309814},{"id":"https://openalex.org/C2777019822","wikidata":"https://www.wikidata.org/wiki/Q390066","display_name":"Containment (computer programming)","level":2,"score":0.30869999527931213},{"id":"https://openalex.org/C181789720","wikidata":"https://www.wikidata.org/wiki/Q4812191","display_name":"Asymptotically optimal algorithm","level":2,"score":0.3041999936103821},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.2888999879360199},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.2874999940395355},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.2766000032424927},{"id":"https://openalex.org/C157125643","wikidata":"https://www.wikidata.org/wiki/Q884707","display_name":"Block code","level":3,"score":0.25459998846054077},{"id":"https://openalex.org/C2400350","wikidata":"https://www.wikidata.org/wiki/Q1752667","display_name":"Linear code","level":4,"score":0.25429999828338623}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.18323","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.18323","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,"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.2605.18323","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.18323","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,"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":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"The":[0],"finite-length":[1],"performance":[2],"of":[3,149,178],"spatially-coupled":[4],"low-density":[5],"parity-check":[6],"(SC-LDPC)":[7],"codes":[8],"is":[9,192],"strongly":[10],"affected":[11],"by":[12,21,43],"short":[13],"cycle":[14],"configurations":[15],"and":[16,63,116,118,164],"the":[17,31,35,39,44,79,99,104,139,147,179],"harmful":[18,91],"structures":[19],"induced":[20],"them.":[22],"This":[23],"paper":[24],"studies":[25],"SC-LDPC":[26,185],"code":[27],"design":[28,36,181],"directly":[29],"at":[30,168],"protograph":[32],"level,":[33],"where":[34],"variables":[37],"are":[38,123],"edge-spreading":[40,151],"assignments":[41,159],"specified":[42],"partition":[45],"matrix.":[46],"In":[47],"contrast":[48],"to":[49,82,107,127,142],"CLLL/Moser--Tardos":[50],"based":[51],"constructive":[52],"frameworks":[53],"for":[54,87,119,158,183],"QC-SC-LDPC":[55],"codes,":[56,186],"we":[57,77],"focus":[58],"on":[59,146],"sharper":[60],"nonconstructive":[61],"existence":[62],"counting":[64,156],"bounds.":[65,135],"By":[66],"encoding":[67],"cycle-activation":[68],"conditions":[69,86],"as":[70,111,113],"polynomial":[71],"vanishing":[72],"constraints":[73],"over":[74],"finite":[75],"grids,":[76],"apply":[78,138],"Combinatorial":[80],"Nullstellensatz":[81],"derive":[83],"sufficient":[84],"memory":[85,105],"eliminating":[88],"prescribed":[89],"cycle-induced":[90],"structures.":[92],"For":[93],"fully":[94],"connected":[95],"$(\u03b3,\u03ba)$":[96],"base":[97],"graphs,":[98],"resulting":[100],"bounds":[101,145],"explicitly":[102],"characterize":[103],"required":[106],"destroy":[108],"all":[109,114,162],"$4$-cycles":[110,163],"well":[112],"$4$-":[115],"$6$-cycles,":[117],"fixed":[120],"$\u03b3$,":[121],"they":[122],"asymptotically":[124],"tight":[125],"up":[126],"a":[128,174],"constant":[129],"factor":[130],"compared":[131],"with":[132],"known":[133],"lower":[134,144],"We":[136],"further":[137],"Alon--F\u00fcredi":[140],"theorem":[141],"obtain":[143],"number":[148],"feasible":[150,180],"assignments,":[152],"including":[153],"an":[154],"explicit":[155],"bound":[157],"that":[160],"eliminate":[161],"hence":[165],"yield":[166],"girth":[167],"least":[169],"six.":[170],"These":[171],"results":[172],"provide":[173],"refined":[175],"algebraic-combinatorial":[176],"characterization":[177],"space":[182],"high-memory":[184],"although":[187],"no":[188],"corresponding":[189],"construction":[190],"algorithm":[191],"provided.":[193]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-05-20T00:00:00"}
