{
  "abstract": "Proximity gaps and correlated agreement underpin the soundness of interactive oracle proofs. We establish these properties for general one-point algebraic geometry (AG) codes, extending corresponding results for Reed-Solomon codes. For codes of length $n$ over $\\mathbb F_q$ with designed distance $n-m$, write $ρ=m/n$ and fix $ρ\\in(0,1)$. For polynomial curves of degree at most $\\ell$ and any $0<\\varepsilon<1$, we prove correlated agreement with multiplicative slack $\\varepsilon$ and soundness $O(\\ell(\\ell+1)/(\\varepsilon q))$ whenever $e\\le(1-\\varepsilon)(n-m)/2$, with no additional genus penalty. This gives asymptotically good AG code families from Garcia--Stichtenoth tower of unbounded length over a fixed field with soundness bounds independent of $n$, beyond the RS restriction $n\\le q$. We also establish mutual correlated agreement up to radius $(1-\\sqrtρ-η)n$, with soundness $O(\\ell n/(η^3q))$ for $0<η<1-\\sqrtρ$. Our half designed distance proof combines auxiliary divisors, simultaneous Berlekamp-Welch interpolation, and error-incidence counting to obtain a common polynomial representation without a Polishchuk-Spielman divisibility lemma. Finally, two counterexamples on the Garcia-Stichtenoth tower show limitations beyond half the minimum distance, at radii below the Johnson bound and with positive slack. The first requires a positive constant soundness lower bound independent of $q$ and $n$, precluding a uniform $O(1/q)$ guarantee throughout the Johnson regime. The second fixes the field and lets the relative gap above half distance tend to zero while ruling out correlated agreement for every soundness parameter $σ<1$, with positive limiting rate and relative error radius.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39256",
  "author_total": 4,
  "authors": [
    {
      "name": "Songsong Li"
    },
    {
      "name": "Chaoping Xing"
    },
    {
      "name": "Chen Yuan"
    },
    {
      "name": "Ruiqi Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/av51p6k9ywhjt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39256",
  "fulltext_url": null,
  "id": "av51p6k9ywhjt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39256",
  "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.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39256",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/av51p6k9ywhjt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39256",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39256"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "On Proximity Gaps for One-Point Algebraic Geometric Codes",
  "title_text": "On Proximity Gaps for One-Point Algebraic Geometric Codes",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/av51p6k9ywhjt",
  "venue": "arXiv",
  "versions": []
}