On Almost Perfect Linear Lee Codes of Packing Radius 2

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
$r\geq 2$
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.