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 \(n\rightarrow \infty \).

基于\({\mathbb {R}}^n\)中交叉多胞体的堆积密度，50多年前Golomb和Welch证明了\({\mathbb {Z}}中Lee球体的堆积密度^n\)必须严格小于 1，前提是 Lee 球体的半径r与n相比足够大，这意味着对应参数r和n不存在完美的 Lee 代码。在本文中，我们研究了对于无限多个n具有固定半径r的 Lee 球体的晶格堆积密度。首先，我们提出了一种方法来验证半径为 2 的 Lee 球的第二密集晶格堆积的不存在性。其次，我们考虑密度为\(\delta _n\rightarrow \frac{2^r}{(2r +1)r!}\)为\(n\rightarrow \infty \)。当\(r=2\)时，堆积密度可以提高到\(\delta _n\rightarrow \frac{2}{3}\)为\(n\rightarrow \infty \)。