{
  "abstract": "The Tu-Deng Conjecture, made in 2011, asserts that for $k\\geq2$ and $1\\leq t<2^k-1$, the set \\[ S_{t,k}=\\left\\{(a,b)\\in\\{0,1,\\ldots,2^k-2\\}^2: a+b\\equiv t\\pmod{2^k-1},\\quad \\wt(a)+\\wt(b)<k\\right\\} \\] satisfies \\[ |S_{t,k}|\\leq 2^{k-1}. \\] Proofs of the conjecture were given independently by the author and by Liu, Luo and Xie in arXiv in 2026. Let $r=\\wt(t)$, and let $z=k-r$ be the number of zeros in the $k$-bit expansion of $t$. We prove that equality holds in the Tu-Deng Conjecture if and only if every cyclic distance between consecutive zeros is at least $z$. We give several other simple necessary and sufficient conditions, including a forbidden-pattern condition, a packing condition for the zero positions, a composition condition for the gaps, and equivalent conditions involving the coefficients of the polynomial $H_W(u,v)$ in the proof of the Tu-Deng conjecture. We also count the equality words having a prescribed number of zeros.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12630",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas W. Cusick"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b79zdysr6rqcq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12630",
  "fulltext_url": null,
  "id": "b79zdysr6rqcq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12630",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7221515000",
  "pdf_url": "https://arxiv.org/pdf/2610.12630",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b79zdysr6rqcq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12630",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12630"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Equality Cases in the Tu-Deng Conjecture Bound",
  "title_text": "Equality Cases in the Tu-Deng Conjecture Bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/b79zdysr6rqcq",
  "venue": "arXiv",
  "versions": []
}