It is shown that if the automorphism group of a binary self-dual code satisfies a certain condition then the code contains words of weight congruent to2modulo4. In particular, no cyclic binary self-dual code can have all its weights divisible by four. The number of cyclic binary self-dual codes of length n is determined, and the shortest nontrivial code in this class is shown to have length14.

It is an interesting open question whether an extremal (72, 36, 16) doubly even code C exists. In [3] the odd prime numbers which can divide the order of the group of C were determined. The largest of these, 23, was eliminated by finding weight 12 vectors in 384 codes [8]. The next largest prime remaining is 17. It is shown that 17 is also not possible by reducing the problem to the consideration of16.17^{3}codes and then finding a weight 12 vector, by computer, in each of these codes.