{
  "abstract": "Let $A_q(n,d)$ denote the largest size of a code of length $n$ and minimum Hamming distance $d$ over an alphabet of $q$ symbols. For fixed $q\\ge2$ and $d\\ge4$ we construct codes with redundancy $n-\\log_q A_q(n,d)\\le(d-3)\\log_q n+O(\\sqrt{\\log n})$ for all large $n$. No algebraic structure is imposed on the alphabet, and the codes are not linear. For $d=4$ and $d=5$ the coefficient of $\\log_q n$ matches the Hamming bound, so the redundancy is determined up to lower-order terms. At distance four the codes are provably better than every linear code: by the polynomial-method bound for cap sets, $\\mathbb{F}_q$-linear codes with $q>2$ require a strictly larger coefficient in front of $\\log_q n$. The construction imposes moment checksums over an auxiliary finite field, as BCH codes do, but one fewer than the distance would normally require. The affine-line theorem from prior work describes the dependencies left by these checks. We exclude them by placing the code positions inside a specially structured subset of the field, built from Behrend's sphere construction with the help of algebraic number theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19960",
  "author_total": 1,
  "authors": [
    {
      "name": "Sergey Yekhanin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2hs8407s8etty.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19960",
  "fulltext_url": null,
  "id": "2hs8407s8etty",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19960",
  "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.DM"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7222054549",
  "pdf_url": "https://arxiv.org/pdf/2610.19960",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2hs8407s8etty.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19960",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19960"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Nonlinear fixed-distance codes via real lifts of moment configurations",
  "title_text": "Nonlinear fixed-distance codes via real lifts of moment configurations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/2hs8407s8etty",
  "venue": "arXiv",
  "versions": []
}