A classification of all four-circulant extremal codes of length 32 over $\Bbb F_2+u\Bbb F_2$ is done by using four-circulant binary self-dual codes of length 32 of minimum weights 6 and 8. As Gray images of these codes, a substantial number of extremal binary self-dual codes of length 64 are obtained. In particular a new code with $\beta=80$ in $W_{64,2}$ is found. Then applying an extension method from the literature to extremal self-dual codes of length 64, we have found many extremal binary self-dual codes of length 66. Among those, five of them are new codes in the sense that codes with these weight enumerators are constructed for the first time. These codes have the values $\beta=1,30,34,84,94$ in $W_{66,1}$.