Automorphisms of codes with applications to extremal doubly even codes of length 48

Abstract

General results on automorphisms of self-dual binary codes are given. These results are applied to the study of extremal self-dual doubly even binary codes of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">48</tex> . The main theorem proved is that an extremal self-dual doubly even code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">48</tex> with a nontrivial automorphism of odd order is equivalent to the extended quadratic residue code. Interesting constructions of the binary extended Golay code as well as a conjecture about a possible connection between an extremal self-dual doubly even code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">72</tex> and an extremal quaternary code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">24</tex> arc yielded by techniques used in the proof.