  In this paper we investigate the spectrum of super-simple 2-$(v,5,1)$
directed designs (or simply super-simple 2-$(v,5,1)$DDs) and also the size of
their smallest defining sets.
  We show that for all $v\equiv1,5\ ({\rm mod}\ 10)$ except $v=5,15$ there
exists a super-simple $(v,5,1)DD$. Also for these parameters, except possibly
$v=11,91$, there exists a super-simple 2-$(v,5,1)$DD whose smallest defining
sets have at least a half of the blocks.
