Self-Dual Codes and Modules for Finite Groups in Characteristic Two

Abstract

Using representation theoretical methods we investigate self-dual group codes and their extensions in characteristic 2. We prove that the existence of a self-dual extended group code heavily depends on a particular structure of the group algebra KG which can be checked by an easy-to-handle criteria in elementary number theory. Surprisingly, in the binary case such a code is doubly even if the converse of Gleason's theorem holds true, i.e., the length of the code is divisible by 8. Furthermore, we give a short representation theoretical proof of an earlier result of Sloane and Thompson which states that a binary self-dual group code is never doubly even if the Sylow 2-subgroups of G are cyclic. It turns out that exactly in the case of a cyclic or Klein four group as Sylow 2-subgroup doubly even group codes do not exist.