Self-dual codes with an automorphism of order 7 and s-extremal codes of length 68

This paper studies and classifies optimal binary self-dual codes having an automorphism of order 7 with 9 cycles. This classification is done by applying a method for constructing binary self-dual codes with an automorphism of odd prime order p. There are exactly 69781 inequivalent binary self-dual [64,32,12] codes with an automorphism of type 7−(9,1). As for binary [66,33,12] self-dual codes with an automorphism of type 7−(9,3) there are 1652432 such codes. We also construct more than 4 million new optimal codes of length 68 among which are the first known examples of the very elusive s-extremal self-dual codes. We prove the nonexistence of [70,35,14] codes with an automorphism of type 7−(9,7). Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before.