The group of permutational automorphisms of a $q$-ary Hamming code

We prove that the group of permutation automorphism of a $q$-ary Hamming code of length $n=(q^m-1)/(q-1)$ is isomorphic to the unitriangular group $\mathbf{UT}_m(q)$ if the code has a parity-check matrix composed of all columns of the form $(0\dots0\,1*\dots*)^\mathsf T$. We also show that the group of permutation automorphisms of a cyclic Hamming code cannot be isomorphic to $\mathbf{UT}_m(q)$. We thus show that equivalent codes can have different permutation automorphism groups.