On extremal codes with automorphisms

Let C be a binary extremal self-dual doubly-even code of length n ≥ 48. If such a code has an automorphism σ of prime order p ≥ 5 then the number of fixed points in the permutation action on the coordinates is bounded by the number of p-cycles. It turns out that large primes p, i.e. n − p small, occur extremely rarely in Aut(C). Examples are the extended quadratic residue codes. We prove that doubly-even extended quadratic residue codes of length n = p + 1 are extremal if and only if n = 8, 24, 32, 48, 80 or 104. Moreover, we reduce the list of putative extremal doubly-even codes with an automorphism of prime order p = n − 1 to merely 12 cases. We conjecture that in fact such an extremal code, if it is not an extended quadratic residue code of one of the lengths given above, does not exist.