Super-simple (v,4,2)(v,4,2) directed designs and a lower bound for the minimum size of their defining set

In this paper, we show that for all v≡1v≡1 (mod 3), there exists a super-simple (v,4,2)(v,4,2) directed design. Also, we show that for these parameters there exists a super-simple (v,4,2)(v,4,2) directed design whose each defining set has at least a half of the blocks.