  In this work, we propose a modified four circulant construction for self-dual
codes and a bordered version of the construction using the properties of
\lambda-circulant and \lambda-reverse circulant matrices. By using the
constructions on $F_2$, we obtain new binary codes of lengths 64 and 68. We
also apply the constructions to the ring $R_2$ and considering the $F_2$ and
$R_1$-extensions, we obtain new singly-even extremal binary self-dual codes of
lengths 66 and 68. More precisely, we find 3 new codes of length 64, 15 new
codes of length 66 and 22 new codes of length 68. These codes all have weight
enumerators with parameters that were not known to exist in the literature.
