On the structure of binary self-dual codes having an automorphism of order a square of an odd prime

We describe a method for constructing binary self-dual codes having an automorphism of order p² for an odd prime p. Using this method, we classify the optimal self-dual codes of lengths 36 ≤ n ≤ 44 and n = 54, having an automorphism of order 9. We obtain all self-dual (56,28,12),(58,29,10), and (60,30,12) codes having an automorphism of order 9 without cycles of length 3. Some of the constructed codes of lengths 54,58, and 60 have weight enumerators for which the existence of codes was not known before.