On the Existence of Complete Coverings of Lee Spheres
Abstract
Every perfect r-error-correcting Lee code in ℤn 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≥ 8 and r2+⌊ 25r/34⌋−3≥ 2n. For the corresponding Cartesian products of cycles, this strengthens Špacapan's nonexistence theorem by replacing the linear radius condition r≥ n with a threshold of order √(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 ε>0, no complete covering exists when n≥ 3, r is sufficiently large, and r2+(2/e−ε)r≥ 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.
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BibTeX
@misc{zhang2026existence,
title = {{On the Existence of Complete Coverings of Lee Spheres}},
author = {Jing Zhang and Yue Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34805},
eprint = {2610.34805},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34805},
}
RIS
TY - UNPB
TI - On the Existence of Complete Coverings of Lee Spheres
AU - Jing Zhang
AU - Yue Zhou
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34805
UR - https://arxiv.org/abs/2610.34805
AB - Every perfect r-error-correcting Lee code in 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≥ 8 and r^2+⌊ 25r/34⌋−3≥ 2n. For the corresponding Cartesian products of cycles, this strengthens Špacapan's nonexistence theorem by replacing the linear radius condition r≥ n with a threshold of order √(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 ε>0, no complete covering exists when n≥ 3, r is sufficiently large, and r^2+(2/e−ε)r≥ 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.
ER -