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

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