  We investigate perfect codes in $\mathbb{Z}^n$ under the $\ell_p$ metric.
Upper bounds for the packing radius $r$ of a linear perfect code, in terms of
the metric parameter $p$ and the dimension $n$ are derived. For $p = 2$ and $n
= 2, 3$, we determine all radii for which there are linear perfect codes. The
non-existence results for codes in $\mathbb{Z}^n$ presented here imply
non-existence results for codes over finite alphabets $\mathbb{Z}_q$, when the
alphabet size is large enough, and has implications on some recent
constructions of spherical codes.
