On non-full-rank perfect codes over finite fields

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 − 1)/(q − 1) is a q-ary constant-weight code with Hamming weight equals to qm−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.