Automorphisms of Order $2p$ in Binary Self-Dual Extremal Codes of Length a Multiple of 24
Martino Borello, Wolfgang M. Willems · IEEE Transactions on Information Theory · 2013
Let$C$be a binary self-dual code with an automorphism$g$of order$2p$, where$p$is an odd prime, such that$g^{p}$is a fixed point free involution. If$C$is extremal of length a multiple of 24, all the involutions are fixed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code$C$with coding theoretical ones of the subcode$C(g^{p})$which consists of the set of fixed points of$g^{p}$, we prove that$C$is a projective$ {\BBF }_{2}\langle g \rangle $-module if and only if a natural projection of$C(g^{p})$is a self-dual code. We then discuss easy-to-handle criteria to decide if$C$is projective or not. As an application, we consider in the last part extremal self-dual codes of length 120, proving that their automorphism group does not contain elements of order 38 and 58.