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

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A classical result of Conway and Pless is that a natural projection of the fixed 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 fixed point free automorphism of order 2. If m is odd or if m=2k with (5k−1k−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.