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

Abstract

Let 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{minus 45 degree rule}{103, 111, 115, 119, 121, 122, 123}⊆J[16], and Iυ{minus 45 degree rule} {qυ-h:h=17, 18, 19, 21, 25}⊆J[υ] for all υ=2n, n≥5. © 1981.