{
  "abstract": "For an integer $q\\ge5$, let $A_q(n,\\{2,4\\})$ be the largest cardinality of a subset of a $q$-ary Hamming space of length $n$ in which every two distinct words have distance $2$ or $4$. Set \\[ N(q)=\\left\\lceil\\max\\left\\{2q+1, \\frac{17+\\sqrt{8q^2+96q-95}}2\\right\\}\\right\\rceil. \\] We prove that $A_q(n,\\{2,4\\})=1+\\binom n2$ for every $n\\ge N(q)$. In particular, $N(q)=2q+1$ for $q\\ge26$. An explicit code of size $q(n-2)$ shows that the least eventual threshold $N_*(q)$ satisfies $N_*(q)\\ge2q-1$; consequently $2q-1\\le N_*(q)\\le2q+1$ for every $q\\ge26$. The argument applies to arbitrary integer alphabet sizes, without a linearity assumption. Its main ingredients are synchronization of symbols on intersecting supports, weighted bounds for $2$-intersecting families of four-sets, and a joint estimate for the weight-two and weight-four layers in a common-three-core configuration. No computational verification is required.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30613",
  "author_total": 3,
  "authors": [
    {
      "name": "Valjakas Djaljapayan"
    },
    {
      "name": "Yen-Chi Roger Lin"
    },
    {
      "name": "Wei-Hsuan Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2z7b2wrknh7jh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30613",
  "fulltext_url": null,
  "id": "2z7b2wrknh7jh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30613",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223383671",
  "pdf_url": "https://arxiv.org/pdf/2610.30613",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2z7b2wrknh7jh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30613",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30613"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Asymptotically Sharp Length Thresholds for \\texorpdfstring{$q$}{q}-ary Codes with Distances \\texorpdfstring{$2$}{2} and \\texorpdfstring{$4$}{4}",
  "title_text": "Asymptotically Sharp Length Thresholds for \\texorpdfstring{q}{q}-ary Codes with Distances \\texorpdfstring{$22 and texorpdfstring4$}{4}",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/2z7b2wrknh7jh",
  "venue": "arXiv",
  "versions": []
}