New extremal singly even self-dual codes of lengths 64 and 66

Abstract

For lengths $64$ and $66$, we construct six and seven extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist, respectively. We also construct new $40$ inequivalent extremal doubly even self-dual $[64,32,12]$ codes with covering radius $12$ meeting the Delsarte bound. These new codes are constructed by considering four-circulant codes along with their neighbors and shadows.