On involutions in extremal self-dual codes and the dual distance of semi self-dual codes

A classical result of Conway and Pless is that a natural projection of the xed code of an automorphism of odd prime order of a self-dual binary linear code is self-dual [13]. In this paper we prove that the same holds for involutions under some (quite strong) conditions on the codes. In order to prove it, we introduce a new family of binary codes: the semi self-dual codes. A binary self-orthogonal code is called semi self-dual if it contains the all-ones vector and is of codimension 2 in its dual code. We prove upper bounds on the dual distance of semi self-dual codes. As an application we get the following: let C be an extremal self-dual binary linear code of length 24m and σ ∈ Aut(C) be a xed point free automorphism of order 2. If m is odd or if m = 2k with ( 5k−1 k−1 ) odd then C is a free F2〈σ〉module. This result has quite strong consequences on the structure of the automorphism group of such codes.