On Lifting Perfect Codes

In this paper, we consider completely regular codes, obtained from perfect (Hamming) codes by lifting the ground field. More exactly, for a given Hamming code
$C$
of length
$n=(q^m-1)/(q-1)$
over
${\BBF}_q$
with a parity check matrix
$H_m$
, we define a new linear code
$C_{(m,r)}$
of length
$n$
over
${\BBF}_{q^r},$

$r \geq 2$
, with this parity check matrix
$H_m$
. The resulting code
$C_{(m,r)}$
is completely regular with covering radius
$\rho = \min\{r,m\}$
. We compute the intersection numbers of such codes and we prove that Hamming codes are the only codes that, after lifting the ground field, result in completely regular codes. Finally, we also prove that extended perfect (Hamming) codes, for the case when extension increases their minimum distance, are the only codes that, after lifting the ground field, result in uniformly packed (in the wide sense) codes.