Smallest defining sets of super-simple 2 - (v, 4,1) directed designs

A $2-(v,k,λ)$ directed design (or simply a $2-(v,k,λ)DD$) is super-simple if its underlying $2-(v,k,2λ)BIBD$ is super-simple, that is, any two blocks of the $BIBD$ intersect in at most two points. A $2-(v,k,λ)DD$ is simple if its underlying $2-(v,k,2λ)BIBD$ is simple, that is, it has no repeated blocks. A set of blocks which is a subset of a unique $2-(v,k,λ)DD$ is said to be a defining set of the directed design. A smallest defining set, is a defining set which has smallest cardinality. In this paper simultaneously we show that the necessary and sufficient condition for the existence of a super-simple $2-(v,4,1)DD$ is $v\equiv1\ ({\rm mod}\ 3)$ and for these values except $v=7$, there exists a super-simple $2-(v,4,1)DD$ whose smallest defining sets have at least a half of the blocks. And also for all $ε> 0$ there exists $v_0(ε)$ such that for all admissible $v>v_0$ there exists a $2-(v,4,1)DD$ whose smallest defining sets have at least $(5/8-\frac{c}{v})\mid \mathcal{B}\mid$ blocks, for suitable positive constant c.

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