{
  "abstract": "We study algebraic geometry codes on hyperelliptic curves of genus $g \\geq 2$ equipped with a rational Weierstrass point, with a focus on complementarity properties. Our first contribution is a characterization of non-special divisors of degree $g$ and $g-1$ via the polynomial degrees of their reduced Mumford representation, reducing a classical geometric problem to a degree test on univariate polynomials. Using this, we construct Linear Complementary Pairs (LCP) of codes via polynomial arithmetic on the Jacobian and provide a criterion in terms of Mumford degrees for the resulting codes to be Maximum Distance Separable (MDS). Under a $2$-torsion condition in the Jacobian, equivalently a divisibility condition on the Mumford polynomials, we obtain explicit multipliers that turn these pairs into Linear Complementary Dual (LCD) codes. We give an MDS LCP with parameters $[8,4,5]_{16}$ in characteristic two, an MDS LCD code with parameters $[10,5,6]_{25}$ on the maximal hyperelliptic curve $y^2=x^5+x$, and an LCD code with parameters $[28,14]_{49}$ on $y^2=x^7+x$. Motivated by the explicit $q=5$ computation, we conjecture for odd prime powers $q\\ge5$ the existence of MDS LCD codes with parameters $[2q,q,q+1]_{q^2}$ arising from this construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.16945",
  "author_total": 3,
  "authors": [
    {
      "name": "Adler Marques"
    },
    {
      "name": "Yuri da Silva"
    },
    {
      "name": "Saeed Tafazolian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/71yvz6sf6e4f2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.16945",
  "fulltext_url": null,
  "id": "71yvz6sf6e4f2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.16945",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7169848423",
  "pdf_url": "https://arxiv.org/pdf/2607.16945",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/71yvz6sf6e4f2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2607.16945",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.16945"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation",
  "title_text": "Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/71yvz6sf6e4f2",
  "venue": "arXiv",
  "versions": []
}