An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes I

Abstract

We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F2+uF2 and F4+uF4; using groups of order 4 and 8. Through these constructions and their extensions, we find binary self-dual codes of lengths 16, 32, 48, 64 and 68, many of which are extremal. In particular, we find forty new extremal binary self-dual codes of length 68, including twelve new codes with γ=5 in W68,2, which is the first instance of such a γ value in the literature.