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

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