On the packing density of Lee spheres

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 \(n\rightarrow \infty \).