A restriction on the weight enumerator of a self-dual code

Gleason's theorem gives the general form of the weight enumerator of a linear binary self-dual code; it is a linear combination with integral coefficients of certain polynomials. When the subcode of words whose weights are multiples of 4 is not the whole code, the MacWilliams identities applied to that subcode yield divisibility conditions on those coefficients. The conditions show that there are no further extremal codes, for Case 1 in the sense of Mallows and Sloane, than the ones known.