On the Non-Existence of Perfect Codes in the Niederreiter-Rosenbloom-Tsfasman Metric

In this paper we consider codes in
$\mathbb F_{q}^{s\times r}$
with packing radius
$R$
regarding the NRT-metric (i.e. when the underlying poset is a disjoint union of
$s$
chains with the same length
$r$
) and we establish necessary condition on the parameters
$s,r$
and
$R$
for the existence of perfect codes. More explicitly, for
$r,s\geq 2$
and
$R\geq 1$
we prove that if there is a non-trivial perfect code then
$(r+1)(R+1)\leq rs$
. We also establish a correspondence between perfect codes with
$r > R$
and those with
$r=R$
. Using this correspondence we prove the non-existence of non-trivial perfect codes in the cases
$s\leq R+2$
and
$s=3$
over non-binary alphabet.