Classification of Extremal Self-Dual Quaternary Codes of Lengths 30 and 32

We classify extremal Hermitian self-dual quaternary codes of lengths 30 and 32 with an automorphism of odd prime order. We prove that there exists exactly one extremal Hermitian self-dual quaternary code with a nontrivial automorphism of odd prime order, up to equivalence; the order of its automorphism group is 36 540. In fact, this code is equivalent to the extended quadratic residue code. We also prove that there exists no extremal Hermitian self-dual quaternary code with a nontrivial automorphism of odd prime order.