An improved recursive construction for disjoint Steiner quadruple systems

Let D(n) be the number of pairwise disjoint Steiner quadruple systems (SQS) of order n. A simple counting argument shows that D(n) <= 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 equivalent to 2 or 4 (mod 6) is large enough. When n >= 7 and n equivalent to 1 or 5 (mod 6), we present a recursive construction and prove a recursive formula on D (4n), as follows: D(4n) >= 2n + min{D(2n), 2n-7}. The related construction has a few advantages over some of the previously known constructions for pairwise disjoint SQSs.