{
  "abstract": "We study the Hamming weight enumerator of the projective Reed-Muller code $\\mathcal C_{2,4}$ over $\\mathbf F_q$, with $\\operatorname{char}(\\mathbf F_q)>3$. We determine the geometrically reducible and singular contributions by combining incidence counts with counts of plane linear series on a fixed normalization. The line and conic configurations and the genus zero cases give explicit polynomials. The line-cubic and genus one cases give formulas involving class numbers. The genus two contribution is a finite sum over hyperelliptic models, using their point counts over $\\mathbf F_q$ and $\\mathbf F_{q^2}$ and their rational Weierstrass points. The remaining smooth contribution is the weighted point count distribution of nonhyperelliptic genus three curves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27392",
  "author_total": 1,
  "authors": [
    {
      "name": "Xu Zhuang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9y0n8p0hvqzq8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27392",
  "fulltext_url": null,
  "id": "9y0n8p0hvqzq8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27392",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223093420",
  "pdf_url": "https://arxiv.org/pdf/2610.27392",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9y0n8p0hvqzq8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27392",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27392"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Projective Reed-Muller Code from Quartic Curves in the Projective Plane",
  "title_text": "The Projective Reed-Muller Code from Quartic Curves in the Projective Plane",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/9y0n8p0hvqzq8",
  "venue": "arXiv",
  "versions": []
}