{
  "abstract": "We study the least pole order of a rational function whose nonzero evaluations scale a prescribed code to a Euclidean LCD code. A residue-one differential yields an obstruction for odd-dimensional one-point AG codes in characteristic two: for $C=C_L(D,aQ)$ with $0\\le a<n$ and odd dimension, the threshold is at least $n+2g-a-1$. The obstruction uses alternating Gram matrices and does not require the scaled code to be self-orthogonal. For the dimension-three codes $C_L(D,(r+1)Q)$ on Hermitian curves over $\\F_{r^2}$, $r=2^e\\ge4$, we obtain \\[ n+2g-r-2\\leτ_D(C;Q)\\le n+2g-3. \\] Consequently the leading genus contribution is asymptotically sharp even in dimension three. On the elliptic curve $y^2+y=x^3$ over $\\F_4$, we prove the exact dimension-three threshold $6$ by an explicit multiplier. For repetition codes on the Hermitian family, the threshold equals the general interpolation bound $n+2g-1$. The upper bounds use information sets and hold for $q>3k$, and for $q>2k$ in characteristic two.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04863",
  "author_total": 1,
  "authors": [
    {
      "name": "Rahim Rahmati-Asghar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bh1azscqk1fm7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04863",
  "fulltext_url": null,
  "id": "bh1azscqk1fm7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04863",
  "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/2611.04863",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bh1azscqk1fm7.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04863",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04863"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Geometric Obstructions and Sharp Genus Dependence in LCD Scalings",
  "title_text": "Geometric Obstructions and Sharp Genus Dependence in LCD Scalings",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/bh1azscqk1fm7",
  "venue": "arXiv",
  "versions": []
}