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The Golomb--Welch conjecture for arbitrary centre sets

  • Bocong Chen
  • Jing Huang
  • Gaojun Luo
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Abstract

We prove that for all n≥3 and r≥2, the Lee ball Bn(r) does not tile Zn by translations, resolving the Golomb--Welch conjecture for arbitrary centre sets and completing the existence classification of perfect Lee codes in Zn. The proof combines translation-invariant averaging, boundary contact identities, and integer moment inequalities with coefficient comparisons in stable symmetric functions. Consequently, we obtain finite local obstructions, strict Banach density gaps for packings and coverings, and the nonexistence of exact measurable Lee-ball towers. In the lattice specialization, our result classifies the attainability of the Lee bound for even-degree abelian Cayley graphs, excludes normal topological coverings and Galois graph coverings for Lee-ball completions, and gives a strict node bound for lattice cubature as well as the nonexistence of subgroup-indexed orthogonal exponential bases on Bn(r).

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Cite

BibTeXRIS
BibTeX
@misc{chen2026golombwelch,
  title = {{The Golomb--Welch conjecture for arbitrary centre sets}},
  author = {Bocong Chen and Jing Huang and Gaojun Luo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34291},
  eprint = {2610.34291},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34291},
}
RIS
TY  - UNPB
TI  - The Golomb--Welch conjecture for arbitrary centre sets
AU  - Bocong Chen
AU  - Jing Huang
AU  - Gaojun Luo
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34291
UR  - https://arxiv.org/abs/2610.34291
AB  - We prove that for all n≥3 and r≥2, the Lee ball B_n(r) does not tile Z^n by translations, resolving the Golomb--Welch conjecture for arbitrary centre sets and completing the existence classification of perfect Lee codes in Z^n. The proof combines translation-invariant averaging, boundary contact identities, and integer moment inequalities with coefficient comparisons in stable symmetric functions. Consequently, we obtain finite local obstructions, strict Banach density gaps for packings and coverings, and the nonexistence of exact measurable Lee-ball towers. In the lattice specialization, our result classifies the attainability of the Lee bound for even-degree abelian Cayley graphs, excludes normal topological coverings and Galois graph coverings for Lee-ball completions, and gives a strict node bound for lattice cubature as well as the nonexistence of subgroup-indexed orthogonal exponential bases on B_n(r).
ER  -