The automorphism group of a $q$-ary Hamming code

It is well known that the semilinear symmetry group of a $q$-ary Hamming code $\mathcal H$ with length $n=\frac{q^m-1}{q-1}$ is isomorphic to $\mathit\Gamma L_m(q)$. This does not clarify if all symmetries of the code are semilinear or not. Here we prove that each symmetry of the code constituted by all triples in $\mathcal H$ is semilinear. This implies that every symmetry of the Hamming code is semilinear. So, it is shown that the automorphism group of a $q$-ary Hamming code is isomorphic to the semidirect product $\mathit\Gamma L_m(q)\rightthreetimes\mathcal H$. Bibliogr. 4.