An improved recursive construction for disjoint Steiner quadruple systems

Let $D\left(n\right)$ be the number of pairwise disjoint Steiner quadruple systems (SQS) of order $n$. A simple counting argument shows that $D\left(n\right)\le n?3$ and a set of $n?3$ such systems is called a large set. No nontrivial large set was constructed yet, although it is known that they exist if  $n\equiv 2$ or $4\left(\mathrm{mod}6\right)$ is large enough. When $n\ge 7$ and $n\equiv 1$ or $5\left(\mathrm{mod}6\right)$, we present a recursive construction and prove a recursive formula on $D\left(4n\right)$, as follows: $$D\left(4n\right)\ge 2n+\min \left\{D\left(2n\right),2n?7\right\}.$$ The related construction has a few advantages over some of the previously known constructions for pairwise disjoint SQSs.