The existence of 23 ${2}^{3}$‐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 tλ $t\lambda $ is said to be λt ${\lambda }^{t}$‐decomposable, if its blocks can be partitioned into nonempty collections Bi ${{\rm{ {\mathcal B} }}}_{i}$, 1≤i≤t $1\le i\le t$, such that each Bi ${{\rm{ {\mathcal B} }}}_{i}$ with the point set forms a design with index λ $\lambda $. In this paper, it is proved that there exists a 23 ${2}^{3}$‐decomposable super‐simple (v,4,6) $(v,4,6)$‐BIBD (balanced incomplete block design) if and only if v≥16 $v\ge 16$ and v≡1(mod3) $v\equiv 1(\,\mathrm{mod}\,\,3)$. [ABSTRACT FROM AUTHOR]