  In this work, four circulant and quadratic double circulant (QDC)
constructions are applied to the family of the rings R_k,m. Self-dual binary
codes are obtained as the Gray images of self-dual QDC codes over R_k,m.
Extremal binary self-dual codes of length 64 are obtained as Gray images of
?-four circulant codes over R_2,1 and R_2,2. Extremal binary self-dual codes of
lengths 66 and 68 are constructed by applying extension theorems to the F_2 and
R_2,1 images of these codes. More precisely, 11 new codes of length 66 and 39
new codes of length 68 are discovered. The codes with these weight enumerators
are constructed for the first time in literature. The results are tabulated.
