PreprintarXiv licencearXiv
The Golomb--Welch conjecture for lattice tilings
Abstract
We prove the Golomb--Welch conjecture for lattice tilings: for every n≥3 and r≥2, the integer lattice Zn admits no lattice tiling by Lee balls Bn(r) of radius r. Together with the classical constructions for n≤2 and for r=1, this determines all parameters for which a lattice Lee tiling exists. Equivalently, we show that every abelian Cayley graph of even degree 2n≥6 and diameter r≥2 has strictly fewer than |Bn(r)| vertices.
Related
- The Golomb--Welch conjecture for arbitrary centre sets
- On Lattice Tilings by Symmetric Limited Magnitude Error Balls: Q-polynomials and Ideal Characterization
- On the Lattice Tilings by Lee Balls: Q-Polynomials, Group-Ring Equations, and a Decision Algorithm
- From Sidelnikov-Welch bounds to projection constants
- Counting subspaces with prescribed Welch sums
BibTeX
@misc{chen2026golombwelch,
title = {{The Golomb--Welch conjecture for lattice tilings}},
author = {Bocong Chen and Jing Huang and Gaojun Luo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33983},
eprint = {2610.33983},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33983},
}
RIS
TY - UNPB
TI - The Golomb--Welch conjecture for lattice tilings
AU - Bocong Chen
AU - Jing Huang
AU - Gaojun Luo
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.33983
UR - https://arxiv.org/abs/2610.33983
AB - We prove the Golomb--Welch conjecture for lattice tilings: for every n≥3 and r≥2, the integer lattice Z^n admits no lattice tiling by Lee balls B_n(r) of radius r. Together with the classical constructions for n≤2 and for r=1, this determines all parameters for which a lattice Lee tiling exists. Equivalently, we show that every abelian Cayley graph of even degree 2n≥6 and diameter r≥2 has strictly fewer than |B_n(r)| vertices.
ER -