The Flower Intersection Problem for Steiner Systems \(S(3,4,v), v= 4-2^m, 5-2^m\)

We determine those pairs \((k,v)\), \(v = 4\cdot2^m, 5\cdot2^m\), for which there exists a pair of Steiner quadruple systems on the same \(v\)-set, such that the quadruples in one system containing a particular point are the same as those in the other system and moreover the two systems have exactly \(k\) other quadruples in common.