On q-Ary Shortened-1-Perfect-Like Codes

We study codes with parameters of
$q$
-ary shortened Hamming codes, i.e.,
$(n=(q^{m}-q)/(q-1), q^{n-m}, 3)_{q}$
. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold packings of radius-1 balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal
$q$
-fold packing with minimum distance 2. Secondly, for every admissible length starting from
$n=20$
, we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code.