  The paper deals with the perfect 1-error correcting codes over a finite field
with $q$ elements (briefly $q$-ary 1-perfect codes). We show that the
orthogonal code to the $q$-ary non-full-rank 1-perfect code of length $n =
(q^{m}-1)/(q-1)$ is a $q$-ary constant-weight code with Hamming weight equals
to $q^{m - 1}$ where $m$ is any natural number not less than two. We derive
necessary and sufficient conditions for $q$-ary 1-perfect codes of non-full
rank. We suggest a generalization of the concatenation construction to the
$q$-ary case and construct the ternary 1-perfect codes of length 13 and rank
12.
