{
  "abstract": "A collection of classical codes $C_1,\\dots,C_m$ has product expansion if every low-weight word in the sum of their extensions to the product space can be written as a sum of words supported on few lines, each line being a codeword of the corresponding $C_i$. For $m=2$ product expansion is equivalent to robust testability of the tensor code, while for $m\\ge 3$ it is strictly stronger; it is the hypothesis on the local codes behind recent constructions of good quantum LDPC codes, good quantum locally testable codes, and quantum codes with transversal non-Clifford gates. No explicit family over a fixed field was previously known to achieve constant rate, linear distance for both the code and its dual, and 3-fold product expansion at growing length. We prove that for the one-point evaluation codes $C_i$ of length $N_i=728\\cdot27^{\\,i-1}$ and pole order $\\lfloor N_i/5\\rfloor$ on the Garcia--Stichtenoth tower over $\\mathbb{F}_{729}$, the triple $(C_i,C_i,C_i)$ has product expansion with a constant independent of $i$. The dual codes $C_i^\\perp$ then form an explicit fixed-field family of rate tending to $109/130$ and relative distance at least $8/65$. The proof is geometric: we view the words of the sum code as arrays on the product of three copies of the $i$-th curve of the tower, where the parity checks are sections of a line bundle. An exact generation theorem describes the sum-code words supported on the grid footprint of a suitable divisor. A support-localisation argument then shows that every low-weight array lies on such a divisor, after deleting a small number of coordinates. We also give a sufficient criterion for three-fold product expansion in terms of degree and genus ratios, yielding explicit families over every non-prime finite field of order at least $512$. The principal conclusions have been formally verified in Lean 4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08172",
  "author_total": 2,
  "authors": [
    {
      "name": "Cassie Ding"
    },
    {
      "name": "Min-Hsiu Hsieh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9edf6zrtykmaa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08172",
  "fulltext_url": null,
  "id": "9edf6zrtykmaa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08172",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08172",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9edf6zrtykmaa.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08172",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08172"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Uniform Three-Fold Product Expansion of Algebraic-Geometry Codes over a Fixed Field",
  "title_text": "Uniform Three-Fold Product Expansion of Algebraic-Geometry Codes over a Fixed Field",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/9edf6zrtykmaa",
  "venue": "arXiv",
  "versions": []
}