{"paper":{"title":"Near-Tight Bounds for 3-Query Locally Correctable Binary Linear Codes via Rainbow Cycles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CC","math.CO","math.IT"],"primary_cat":"cs.IT","authors_text":"Omar Alrabiah, Venkatesan Guruswami","submitted_at":"2024-04-08T20:54:14Z","abstract_excerpt":"We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $\\delta > 0$ of adversarial errors must have dimension at most $O_{\\delta}(\\log^2 n \\cdot \\log \\log n)$. This is almost tight in view of quadratic Reed-Muller codes being a $3$-query locally correctable code (LCC) with dimension $\\Theta(\\log^2 n)$. Our result improves, for the binary field case, the $O_{\\delta}(\\log^8 n)$ bound obtained in the recent breakthrough of (Kothari and Manohar, 2023) (arXiv:2311.00558) (and the more recent improvement to $O_{\\delta}(\\log^4 n)$ for bi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.05864","kind":"arxiv","version":1},"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/2404.05864/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"}