{
  "abstract": "We prove that any linear $3$-query locally correctable code $C \\colon \\mathbb{F}_2^k \\to \\mathbb{F}_2^n$ with constant correction radius must have length $n \\geq 2^{Ω(\\sqrt{k})}$. This improves on the prior best lower bound of $2^{Ω(\\sqrt{k/\\log k})}$, due to [AG24], and shows that the folklore construction of linear $3$-LCCs from Reed-Muller codes, which achieve length $n = 2^{O(\\sqrt{k})}$, is optimal up to constant factors in the exponent. Our proof builds on the approach of [KM24a], further refined in [Yan24,KM24b], that uses \"chain derived equations\" in combination with Kikuchi graphs. The crux of our argument is to show that a certain edge-coloured Kikuchi graph has a unique colour cycle: a cycle in which some colour is used exactly once. We do this by controlling the distribution of colours at each vertex in the graph using a random walk mixing argument and a novel hypergraph decomposition based on edge expansion of the hypergraph's \"flattening\" into a graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03145",
  "author_total": 2,
  "authors": [
    {
      "name": "Oliver Janzer"
    },
    {
      "name": "Peter Manohar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e138pfesnspve.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03145",
  "fulltext_url": null,
  "id": "e138pfesnspve",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03145",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03145",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e138pfesnspve.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03145",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03145"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Reed-Muller Codes are Optimal Binary Linear 3-Query LCCs",
  "title_text": "Reed-Muller Codes are Optimal Binary Linear 3-Query LCCs",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/e138pfesnspve",
  "venue": "arXiv",
  "versions": []
}