On the classication of binary self-dual [44, 22, 8] codes with an automorphism of order 3 or 7

Abstract

All binary self-dual [44, 22,8] codes with an automorphism of order 3 or 7 are classified. In this way we complete the classification of extremal self-dual codes of length 44 having an automorphism of odd prime order.

Self-dual codes with automorphism of order 3 having 8 cycles

“…In the first case wt (π −1 (v) + φ −1 (e, e, x 2 e, x 2 e, xe, xe, 0, 0)) = 6, and in the other case we have wt (π −1 (v) + φ −1 (x 2 e, xe, x 2 e, xe, e, 0, e, 0)) = 6. These contradict the minimum weight of C. Therefore wt (v) = 4, or 8 (see [3]). Therefore 0 ≤ k 1 ≤ 2.…”

“…The code π(F σ (C)) is a binary self-dual [22, 11, ≥ 4] code. In this case the code π(F σ (C)) has a generator matrix G π(F) given in (2) where B generates a [8, k 1 , ≥ 4] code, and D generates a self-orthogonal [14, (Bouyukliev [12]) There is only one self-orthogonal [14,3,8] code.…”

“…The unique self-orthogonal [14,3,8] code is the juxtaposition of two copies of the [7,3,4] code, and this code has weight enumerator W = 1 + 7y 8 . [38,19,8] code has an automorphism of order 3 with 8 independent cycles and 14 fixed points.…”

“…The code π(F σ (C)) is a binary self-dual [28, 14, ≥ 4] code. In this case the code π(F σ (C)) has a generator matrix G π(F) given in (2) [3]). In this case D * cannot be self-complementary.…”

“…The classification of length 42 is described in [3]. We apply the method for constructing self-dual codes via an automorphism of odd prime order [5,9,13].…”