On Super-Simple $2$-$(v,4,\lambda )$ Designs

Abstract

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