  More than 50 years ago, Golomb and Welch conjectured that there is no perfect
Lee codes $C$ of packing radius $r$ in $\mathbb{Z}^{n}$ for $r\geq2$ and $n\geq
3$. Recently, Leung and the second author proved that if $C$ is linear, then
the Golomb-Welch conjecture is valid for $r=2$ and $n\geq 3$. In this paper, we
consider the classification of linear Lee codes with the second-best
possibility, that is the density of the lattice packing of $\mathbb{Z}^n$ by
Lee spheres $S(n,r)$ equals $\frac{|S(n,r)|}{|S(n,r)|+1}$. We show that, for
$r=2$ and $n\equiv 0,3,4 \pmod{6}$, this packing density can never be achieved.
