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Are there any strong connections between automorphism groups of codes that are dual codes of each other? I am looking for statements like one charcterizes other or one gives bounds on other etc.

In general are there any instances where charcterization of automorphism groups of code or its dual code lead to showing something of interest for a code?

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    $\begingroup$ Shouldn't they trivially be the same? Permutations preserve inner products. So if you apply a permutation on coordinates of a code and its dual, you get two codes that are duals. Now if the permuted code is the code itself, the permuted dual must also be the dual code. So a permutation preserves the code iff it preserves the dual? $\endgroup$ – Mahdi Cheraghchi Apr 5 '20 at 20:29

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