{
  "abstract": "The containment of the code of the meet $G\\wedge A$ in the hull and the identity $G\\vee A-D=K-G\\wedge A$ exchanging meet and join are known; imposing that $G\\wedge A$ be principal constructs algebraic geometry codes with one-dimensional hull. We turn that construction into a measurement. For arbitrary divisors $G$ and $A$ we compute $C_L(D,G)\\cap C_L(D,A)$ exactly: it is the code of the meet together with an excess $\\varepsilon(G,A)$, canonically their quotient and a subquotient of $H^1(\\mathcal O(G\\wedge A))$. So $\\varepsilon$ vanishes exactly when the meet is non-special; otherwise it certifies that $K-G\\wedge A$ is linearly equivalent to an effective divisor, at degree zero the vanishing of a single class in the Picard group: the hull detects a linear equivalence rather than being built from one. For superelliptic curves $y^n=f(x)$ both sides can be computed: their weighted plane models in $\\mathbb P^2_{(1,n/c,d/c)}$, $c=\\gcd(n,d)$, identify codes $C_s$ of weighted forms of degree $s$ with those of $sD_\\infty$ and turn hulls into lattice counts. The range on which $\\varepsilon$ is blind is an explicit interval of degrees, where $\\dim\\operatorname{Hull}(C_s)=cμ(s)-nδ+1-g_X$, $μ(s)=\\min\\{s,M-s\\}$, depends only on its affine-point count. Outside it the meet and join are invariant under $s\\mapsto M-s$ while $\\varepsilon$ is not, so every asymmetry of the hull profile is excess and the threshold in $s$ refines the divisor class: two totally split curves of genus two, over $\\mathbb F_7$ and over $\\mathbb F_{11}$, present the same class at the same pair of degrees and are separated by the profile alone. If $0\\leq°(G\\wedge A)\\leq2g_X-2$ the hull is at most $g_X+1$, so it is large only where it is blind, and over a prime field, under an explicit inequality on $(n,d,q)$, its maximum over the family is $\\ell(\\lfloor M/2\\rfloor D_\\infty)$, attained exactly on the totally split locus.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02850",
  "author_total": 2,
  "authors": [
    {
      "name": "Jurgen Mezinaj"
    },
    {
      "name": "Tanush Shaska"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/arzprgeh7tbzm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02850",
  "fulltext_url": null,
  "id": "arzprgeh7tbzm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02850",
  "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": "W7172540279",
  "pdf_url": "https://arxiv.org/pdf/2608.02850",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/arzprgeh7tbzm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2608.02850",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02850"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Hulls, linear equivalence, and weighted superelliptic codes",
  "title_text": "Hulls, linear equivalence, and weighted superelliptic codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/arzprgeh7tbzm",
  "venue": "arXiv",
  "versions": []
}