Complexity of finding automorphism group of code
By taking direct sum of codes (given their two generator matrices $G_1, G_2$, consider the block matrix $G_1 \oplus G_2 = \begin{bmatrix} G_1 &amp; 0 \\ 0 &amp; G_2 \end{bmatrix}$), finding the automorphism group is at least as hard as testing isomorphism of codes (also called code equivalence). The current best upper bound for testing equivalence of linear codes is $2^{O(n)}$ (see Babai-Codenotti-G-Qiao, SODA '11), and this algorithm can in fact give you generators for the automorphism group.

(For the above reduction I'm assuming that each code is indecomposable in the sense that neither code is itself a nontrivial direct sum, but I think this is true of (a) most/all codes people ever study and (b) most codes, ie a random code.)