{
  "abstract": "For an integer $q\\ge2$, let $A_q(n,\\{2,4\\})$ be the largest cardinality of an unrestricted $q$-ary code of length $n$ in which any two distinct words have Hamming distance $2$ or $4$. We prove that \\[ A_q(n,\\{2,4\\})=1+\\binom n2 \\qquad(q\\in\\{3,4\\},\\ n\\ge6), \\] thereby settling the two remaining alphabet sizes in the conjecture of Boyvalenkov, Delchev, Zinoviev, and Zinoviev; the binary case was settled previously. The quaternary result implies the ternary result by alphabet inclusion. For $n\\ge16$, the quaternary upper bound is proved combinatorially. After normalizing one word to zero, the weight-four supports form a $2$-intersecting family; exact support intersections force symbol synchronization and restrict support multiplicities. We combine the nontrivial intersection theorem with matching arguments and a three-coordinate projection lemma. The finite lengths $6\\le n\\le13$ are handled by a complete Hamming-isometry reduction to exact clique decisions, and $n=14,15$ by a smaller exact weighted-clique relaxation. All finite decisions are reproduced by two standalone integer-only C++17 verifiers provided as supplementary source files.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30615",
  "author_total": 3,
  "authors": [
    {
      "name": "Yen-Chi Roger Lin"
    },
    {
      "name": "Valjakas Djaljapayan"
    },
    {
      "name": "Wei-Hsuan Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c0fxyje8h1dtr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30615",
  "fulltext_url": null,
  "id": "c0fxyje8h1dtr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30615",
  "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.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223272161",
  "pdf_url": "https://arxiv.org/pdf/2610.30615",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c0fxyje8h1dtr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30615",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30615"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Ternary and Quaternary Cases of the $\\{2,4\\}$-Distance Code Conjecture",
  "title_text": "The Ternary and Quaternary Cases of the {2,4}-Distance Code Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/c0fxyje8h1dtr",
  "venue": "arXiv",
  "versions": []
}