Determination of weight enumerators of binary extremal self-dual [42,21,8] codes

We determine the weight enumerators for which there is a binary extremal self-dual [42,21,8] code. In particular, the number β of the vectors of weight 5 in the shadow satisfies β {0,1,…,22,24,26,28,32,42}. We also give a classification of extremal self-dual [42,21,8] codes with β {24,26,28,32,42}.