An ASQS(v) is a particular Steiner system featuring a set of v vertices and two separate families of blocks, B and G, whose elements have a respective cardinality of 4 and 6. It has the property that any three vertices of X belong either to a B-block or to a G-block. The parameter cb is the number of common blocks in two separate ASQSs, both defined on the same set of vertices X . In this paper it is shown that cb ≤ 29 for any pair of ASQSs(12).