{
  "abstract": "Every perfect $r$-error-correcting Lee code in $\\mathbb{Z}^n$ induces a finite configuration around each codeword: a family of pairwise disjoint radius-$r$ Lee spheres, all disjoint from the central sphere, that covers every lattice point at Lee distance $r+1$ from the codeword. We call such a configuration a complete covering. Under projection onto a suitable Cartesian product of cycles, a complete covering corresponds to an optimal local structure in the sense of Špacapan. The existence of a complete covering is necessary but, in general, not sufficient for the existence of a perfect Lee code, as our construction for $(n,r)=(3,2)$ shows. We prove that no complete covering exists whenever $n\\ge 8$ and $r^2+\\lfloor 25r/34\\rfloor-3\\ge 2n$. For the corresponding Cartesian products of cycles, this strengthens Špacapan's nonexistence theorem by replacing the linear radius condition $r\\ge n$ with a threshold of order $\\sqrt{2n}$. We also obtain an improved large-radius nonexistence bound for perfect Lee codes, strengthening the bound of Horak and Kim. Furthermore, we show that, for every $\\varepsilon>0$, no complete covering exists when $n\\ge 3$, $r$ is sufficiently large, and $r^2+(2/\\mathrm{e}-\\varepsilon)r\\ge 2n$. On the positive side, we use near-optimal ternary constant-weight codes to construct almost complete coverings whose pairwise disjoint shadows cover a $1-o(1)$ proportion of the first outer Lee shell as the dimension tends to infinity, for every fixed radius.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34805",
  "author_total": 2,
  "authors": [
    {
      "name": "Jing Zhang"
    },
    {
      "name": "Yue Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9v2q3zmbt61vq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34805",
  "fulltext_url": null,
  "id": "9v2q3zmbt61vq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34805",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223908960",
  "pdf_url": "https://arxiv.org/pdf/2610.34805",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9v2q3zmbt61vq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34805",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34805"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "On the Existence of Complete Coverings of Lee Spheres",
  "title_text": "On the Existence of Complete Coverings of Lee Spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/9v2q3zmbt61vq",
  "venue": "arXiv",
  "versions": []
}