The existence of 2 3 ${2}^{3}$ -decomposable super-simple ( v , 4 , 6 ) $(v,4,6)$ -BIBDs

A design is said to be super-simple if the intersection of any two blocks has at most two elements. A design with index $t\lambda$ is said to be ${\lambda }^t$-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 there exists a $2^3$-decomposable super-simple $\left(v,4,6\right)$-BIBD (balanced incomplete block design) if and only if $v\ge 16$ and $v\equiv 1\left(\mathrm{mod}3\right)$.