Nonexistence of extremal doubly even self-dual codes with large length

The number of the extremal doubly even self-dual codes is finite (cf. MacWilliams and Sloane, 1977) is based on the fact that A4m+8∗ < 0 for all n ≡ 0 (mod 8) sufficiently large where A4m+8∗ is the coefficient of its extremal weight enumerator polynomial (cf. MacWilliams and Sloane, 1977), m = [n/24]. The present paper shows that A4m+8∗ < 0 for all n ≡ 0 (mod 8) with m ⩾ 166, i.e., there does not exist any extremal doubly even self-dual codes of length n ⩾ 3984. Our result improves the previously known result.