The existence of λ $\lambda $ -decomposable super-simple ( 4 , 2 λ ) $(4,2\lambda )$ -GDDs of type g u ${g}^{u}$ with λ = 2 , 4 $\lambda =2,4$

A design is said to be super-simple if the intersection of any two of its blocks has at most two elements. A design with index $t\lambda$ is said to be $\lambda$-decomposable, if its blocks can be partitioned into nonempty collections ${ℬ}_i$, $1\le i\le t$, such that each ${ℬ}_i$ with the point set forms a design with index $\lambda$. In this paper, it is proved that for $\lambda \in \left\{2,4\right\}$, there exists a $\lambda$-decomposable super-simple $\left(4,2\lambda \right)$-GDD of type $g^u$ if and only if $u\ge 4$, $g\left(u-2\right)\ge 2\lambda$ and $g\left(u-1\right)\equiv 0\left(\mathrm{mod}3\right)$, except for $\left(g,u,\lambda \right)=\left(3,5,2\right)$, and except possibly for $\left(g,u,\lambda \right)\in \left\{\left(2,7,2\right),\left(6,5,4\right)\right\}$.