Almost Perfect Linear Lee Codes of Packing Radius 2 Only Exist for Small Dimensions

It is conjectured by Golomb and Welch around half a century ago that there is no perfect Lee codes
$C$
of packing radius
$r$
in
$\mathbb {Z}^{n}$
for
$r\geq 2$
and
$n\geq 3$
. In 2020, Leung and Zhou proved this conjecture for linear Lee codes with
$r=2$
. A natural question is whether it is possible to classify the second best, i.e., almost perfect linear Lee codes of packing radius 2. Partial results was obtained recently by Xu and Zhou. In this paper, we show that if such codes exist in
$\mathbb {Z}^{n}$
, then
$n$
must belong to
$\{1,2, 11, 29, 47, 56, 67, 79, 104, 121, 134, 191\}$
.