On the nonexistence of extremal self-dual codes

Most of the known nonexistence results for extremal self-dual codes were obtained by means of considering when the corresponding extremal weight enumerators contain negative coefficients. However, some of the known results were based on computer programs and a detailed and unified proof is very much desirable. In the present paper, with a unified notation, we deal with all four types of extremal weight enumerators simultaneously. We show that the third nonzero coefficient in the extremal weight enumerator is negative if and only if for Type I and n = 8i (i ⩾ 4), 8i + 2 (i ⩾ 5), 8i + 4 (i ⩾ 6), 8i + 6 (i ⩾ 7); for Type II and n = 24i (i ⩾ 154), 24i + 8 (i ⩾ 159), 24i + 16 (i ⩾ 164); for Type III and n = 12i (i ⩾ 70), 12i + 4 (i ⩾ 75), 12i + 8 (i ⩾ 78); for Type IV and n = 6i (i ⩾ 17), 6i + 2 (i ⩾ 20), 6i + 4 (i ⩾ 22). Thus the corresponding extremal self-dual codes with length n do not exist.