On binary self-dual extremal codes

There is a large gap between Zhang’s theoretical bound for the length n of a binary extremal self-dual doublyeven code and what we can construct. The largest n is 136. In order to find examples for larger n a non-trivial automorphism group might be helpful. In the list of known examples extended quadratic residue codes and quadratic double circulant codes have large automorphism groups. But in both classes the extremal ones are all known. They are exactly those which are in the list; hence of small length. The investigations we have done so far give some evidence that for larger n the automorphism group of a putative extremal self-dual doubly-even code may be very small, if not trivial. Thus the code merely seems to be a big combinatorial object and therefore possibly hard to find. Due to Mallows-Sloane [7] and Rains [9] a binary selfdual code C of length n and minimum distance d satisfies