  The existence of optimal binary self-dual codes is a long-standing research
problem. In this paper, we present some results concerning the decomposition of
binary self-dual codes with a dihedral automorphism group $D_{2p}$, where $p$
is a prime. These results are applied to construct new self-dual codes with
length $78$ or $116$. We obtain $16$ inequivalent self-dual $[78,39,14]$ codes,
four of which have new weight enumerators. We also show that there are at least
$141$ inequivalent self-dual $[116,58,18]$ codes, most of which are new up to
equivalence. Meanwhile, we give some restrictions on the weight enumerators of
singly even self-dual codes. We use these restrictions to exclude some possible
weight enumerators of self-dual codes with lengths $74$, $76$, $82$, $98$ and
$100$.
