{
  "abstract": "In this paper, we study the lattice tilings of $\\mathbb{Z}^n$ by the symmetric limited magnitude error balls $\\mathcal{B}(n,t,k,k)$, which play a crucial role in flash memory applications. Using generating functions and complete Bell polynomials, we derive an explicit formula for the $r$-th Q-polynomial associated with $\\mathcal{B}(n,t,k,k)$. Then, an ideal-theoretic characterization of the lattice tiling condition is developed: for an odd prime $p$ dividing $|\\mathcal{B}(n,t,k,k)|$ exactly once, a lattice tiling would force a certain polynomial to lie outside a certain ideal of $\\mathbb{F}_p[Y_1,\\dots,Y_{\\frac{p-1}{2}}]$. Applying this criterion yields congruence conditions on the dimension $n$ that rule out lattice tilings; for $2\\le t\\le10$ and $1\\le k\\le10$, a systematic Maple computation over all primes $3\\le p\\le29$ excludes, on average, $35.65\\%$ of all dimensions, with a maximum of $76.69\\%$. Finally, we obtain two infinite families of parameters $(n,t,k)$ for which no lattice tiling exists: no lattice tiling exists if $k\\equiv4\\pmod9$ and the base-$3$ addition of $t$ and $n-1-t$ produces exactly one carry; and no lattice tiling of $\\mathbb{Z}^n$ by $\\mathcal{B}(n,n-1,k,k)$ exists if $k\\equiv8\\pmod9$ and $n\\equiv2,4\\pmod6$. Their sizes are also estimated.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02394",
  "author_total": 1,
  "authors": [
    {
      "name": "Ang Xiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07xehdp5qtprc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02394",
  "fulltext_url": null,
  "id": "07xehdp5qtprc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02394",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02394",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07xehdp5qtprc.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02394",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02394"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On Lattice Tilings by Symmetric Limited Magnitude Error Balls: Q-polynomials and Ideal Characterization",
  "title_text": "On Lattice Tilings by Symmetric Limited Magnitude Error Balls: Q-polynomials and Ideal Characterization",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/07xehdp5qtprc",
  "venue": "arXiv",
  "versions": []
}