Covering Radius of Melas Codes

We prove that the covering radius of the Melas code
$M(m,q)$
of length
$n=q^{m}-1$
over
$\mathbb {F}_{q}$
is 2 if
$q > 3$
. We also prove that the covering radius of
$M(m,3)$
is 3 is
$m \ge 3$
, the covering radius of
$M(2,3)$
is 4, and the covering radii of
$M(1,2)$
and
$M(1,3)$
are 1.