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On the packing density of Lee spheres

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Abstract

Based on the packing density of cross-polytopes in \({\mathbb {R}}^n\), more than 50 years ago Golomb and Welch proved that the packing density of Lee spheres in \({\mathbb {Z}}^n\) must be strictly smaller than 1 provided that the radius r of the Lee sphere is large enough compared with n, which implies that there is no perfect Lee code for the corresponding parameters r and n. In this paper, we investigate the lattice packing density of Lee spheres with fixed radius r for infinitely many n. First we present a method to verify the nonexistence of the second densest lattice packing of Lee spheres of radius 2. Second, we consider the constructions of lattice packings with density \(\delta _n\rightarrow \frac{2^r}{(2r+1)r!}\) as \(n\rightarrow \infty \). When \(r=2\), the packing density can be improved to \(\delta _n\rightarrow \frac{2}{3}\) as \(n\rightarrow \infty \).

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 12371337), the Science and Technology Innovation Program of Hunan Province (No. 2023RC1003) and the Training Program for Excellent Young Innovators of Changsha (No. kq2106006).

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  1. Department of Mathematics, National University of Defense Technology, Changsha, 410073, China

    Ang Xiao & Yue Zhou

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  1. Ang Xiao
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  2. Yue Zhou
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Correspondence to Yue Zhou.

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Xiao, A., Zhou, Y. On the packing density of Lee spheres. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01410-0

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