Automorphisms of Extremal Self-Dual Codes

Let
C
be a binary extremal self-dual code of length
n
¿ 48. We prove that for each
¿ ¿ Aut(C
) of prime order
p
¿ 5 the number of fixed points in the permutation action on the coordinate positions is bounded by the number of
p
-cycles. It turns out that large primes
p
, i.e.,
n
-
p
small, seem to occur in
|Aut(C
)| very rarely. Examples are the extended quadratic residue codes. We further prove that doubly even extended quadratic residue codes of length
n
=
p
+ 1 are extremal only in the cases
n
=8, 24, 32, 48, 80, and 104.