{"paper":{"title":"Exponential Lower Bounds for Smooth 3-LCCs and Sharp Bounds for Designs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Peter Manohar, Pravesh K. Kothari","submitted_at":"2024-04-09T17:59:47Z","abstract_excerpt":"We give improved lower bounds for binary $3$-query locally correctable codes (3-LCCs) $C \\colon \\{0,1\\}^k \\rightarrow \\{0,1\\}^n$. Specifically, we prove:\n  (1) If $C$ is a linear design 3-LCC, then $n \\geq 2^{(1 - o(1))\\sqrt{k} }$. A design 3-LCC has the additional property that the correcting sets for every codeword bit form a perfect matching and every pair of codeword bits is queried an equal number of times across all matchings. Our bound is tight up to a factor $\\sqrt{8}$ in the exponent of $2$, as the best construction of binary $3$-LCCs (obtained by taking Reed-Muller codes on $\\mathbb{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.06513","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.06513/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}