On the automorphism groups of binary linear codes

Let C be a binary linear code and suppose that its automorphism group contains a non trivial subgroup G. What can we say about C knowing G? In this paper we collect some answers to this question in the cases G ∼= Cp, G ∼= C2p and G ∼= D2p (p an odd prime), with a particular regard to the case in which C is self-dual. Furthermore we generalize some methods used in other papers on this subject. Finally we give a short survey on the problem of determining the automorphism group of a putative self-dual [72, 36, 16] code, in order to show where these methods can be applied. This paper is a presentation of some of the main results about the automorphism group of binary linear codes obtained by the author in his Ph.D. thesis. Part of the results are proved in joint papers with Wolfgang Willems, Francesca Dalla Volta and Gabriele Nebe. The problem we want to investigate is the following: let C be a (self-dual) binary linear code and suppose that Aut(C) contains a non trivial subgroup G. What can we say about C knowing G? To face this problem, usually we want to find out “smaller pieces” which are easier to determine and then look at the structure of the whole code. In Section 2 we present a classical decomposition of codes with automorphisms of odd prime order. In Section 3 we summarize the most significant results of [BW], about codes with automorphisms of order 2p, where p is an odd prime. Section 4 is a generalization of methods used in [FN] and [BDN], about codes whose automorphism groups contain particular dihedral groups. Finally, in Section 5 we point out and generalize some theoretical tools used in [Bor1], [BDN] and [Bor2]. Our methods can be applied • to study the possible automorphism groups of extremal self-dual binary linear codes; • to construct self-orthogonal binary linear codes with large minimum distance and relatively large dimension; • to classify self-dual binary linear codes with certain parameters. Obviously the last one is the most ambitious. In the last section, which is a short survey on the problem of determining the automorphism group of a putative extremal self-dual [72, 36, 16] code, we underline where these methods can be applied, showing their power. 2010 Mathematics Subject Classification. Primary 94B05, 20B25. Member INdAM-GNSAGA (Italy), IEEE.