  In this work, quadratic double and quadratic bordered double circulant
constructions are applied to F_4 + uF_4 as well as F_4, as a result of which
extremal binary self-dual codes of length 56 and 64 are obtained. The binary
extension theorems as well as the ring extension version are used to obtain 7
extremal self-dual binary codes of length 58, 24 extremal self-dual binary
codes of length 66 and 29 extremal self-dual binary codes of length 68, all
with new weight enumerators, updating the list of all the known extremal
self-dual codes in the literature.
