Construction of steiner quadruple systems having a prescribed number of blocks in common

AbstractLet qυ=υ(υ–1)(υ–2)/24 and let Iυ={0, 1, 2, …, qυ–14}∪{qυ–12, qυ–8, qυ}, for υ⩾8 Further, let J[υ] denote the set of all k such that there exists a pair of Steiner quadruple systems of order υ having exactly k blocks in common. We determine J[υ] for all υ=2n, n⩾2, with the possible exception of 7 cases for υ=16 and of 5 cases for each υ⩾32. In particular we show: J[υ]⊆Iυ for all υ≡2 or 4 (mod 6) and υ⩾8, J[4]={1}, J[8]=I8={0, 2, 6, 14}, I16⧹{103, 111, 115, 119, 121, 122, 123}⊆J[16], and Iυ⧹ {qυ–h:h=17, 18, 19, 21, 25}⊆J[υ] for all υ=2n, n⩾5