Super-simple $(v,5,2)$ directed designs and their smallest defining sets with its application in LDPC codes

In this paper, we show that for all $v\equiv 0,1$ (mod 5) and $v\ge 15$, there exists a super-simple $(v,5,2)$ directed design, also for these parameters there exists a super-simple $(v,5,2)$ directed design such that its smallest defining sets contain at least half of its blocks. Also, we show that these designs are useful in constructing parity-check matrices of LDPC codes.