{
  "abstract": "Let $S_n$ denote the maximum length of a coil in the $n$-dimensional hypercube $Q_n$, i.e., a simple cycle whose induced subgraph has no additional chords. We combine classical square-counting identities and structural properties of maximum coils with new valid combinatorial inequalities into a parameterized linear program whose optimal value provides an upper bound for $S_n$. For $n\\ge13$, we construct two analytic dual feasible solutions. Their objective values yield upper bounds on $S_n$, and the smaller value is the exact optimum of the parameterized LP. One dual solution results in the bound \\[ S_n\\le 2^{n-1} \\frac{8n(n-1)(5n-14)} {40n^3-131n^2+4n+136-(5n-14)\\lfloor3n/4\\rfloor}, \\] which improves upon the upper bounds in Lukito, Lukito-van Zanten, and Zémor for $13\\le n\\le19853$. The second yields the upper bound \\[ S_n\\le 2^{n-1}\\left( 1-\\frac1{30\\sqrt{3n}}+O(n^{-1}) \\right), \\] improving upon the asymptotic upper bound of Zémor. Taken together, the two bounds improve upon the upper bounds for $S_n$ in Lukito, Lukito-van Zanten, and Zémor for every $n\\ge13$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36461",
  "author_total": 3,
  "authors": [
    {
      "name": "Kevin M. Byrnes"
    },
    {
      "name": "Eric T. Harley"
    },
    {
      "name": "Florin Spinu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6jq4zyjgbaj4a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36461",
  "fulltext_url": null,
  "id": "6jq4zyjgbaj4a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36461",
  "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"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7224110017",
  "pdf_url": "https://arxiv.org/pdf/2610.36461",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6jq4zyjgbaj4a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36461",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36461"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Linear Programming Upper Bounds for the Coil-in-the-Box Problem",
  "title_text": "Linear Programming Upper Bounds for the Coil-in-the-Box Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/6jq4zyjgbaj4a",
  "venue": "arXiv",
  "versions": []
}