It is proven that for all \(v \equiv 1 \mod 3\), \(v \geq 7\) there is a \(2-(v,4,2)\) design whose blocks have pairwise at most two elements in common. Moreover, for \(v \equiv 1, 4 \mod 12\) we have shown that these designs can be generated by two copies of \(2-(v,4,1)\) designs.