Classroom note: Some partitions of S(2, 3, v2) and S(2, 4, 2)

Introduction The aim of this note is to provide a Steiner triple system STS( v ) which admits a partition into v STS( v) for every v 1 or 3 (mod 6). The STS( v ) obtained in this way will also contain ~v( v 1) subsystems of order 3v. An analogous construction is provided of an S(2, 4, v ) with a partition into v S(2, 4, v)'s in the case v 1 (mod 12). The Steiner system constructed here will also contain f2v(v 1) subsystems of order 4v. We observe that the direct product construction of an S(2, 3, v 2 ), an S(2, 4, v 2 ), respectively, provides Steiner systems with a completely different structure. We assume the reader is familiar with the Steiner system terminology and refer to [1, 7) for background and to [5, 6J for more literature on the subject. The constructions 1. S(2, 3, v 2 ) Take any v pairwise disjoint STS(v)'s on the points Aj,Bj, ... ,Zj, j = 1,2 ... v. Their blocks will be blocks of the STS( v 2 ) S we want to construct. Each of the remaining ~V2V( v 1) blocks consists of three points in distinct STS( v )'s. To decide how to form triples of subsystems, we take any ST S( v), Li, on the "points" A, B, . .. , Z. For any block, ABC say, of Li we use Construction 1 in [3], i.e. we construct an STS(3v) having A,B, and C as subsystems. Thus the v 2 blocks coming from the triple ABC are obtained from the v base blocks [3]: