Uniqueness of certain completely regular Hadamard codes

Abstract: We classify binary completely regular codes of length $m$ with minimum distance $\delta$ for $(m,\delta)=(12,6)$ and $(11,5)$. We prove that such codes are unique up to equivalence, and in particular, are equivalent to certain Hadamard codes. We prove that the automorphism groups of these Hadamard codes, modulo the kernel of a particular action, are isomorphic to certain Mathieu groups, from which we prove that completely regular codes with these parameters are necessarily completely transitive.