Some naive constructions of S(2, 3, v) and S(2, 4, v)

Abstract

De Resmini, M.J., Some naive constructions of S(2, 3, u) and S(2, 4, u), Discrete Mathematics 97 (1991) 149-1.53. The aim of this paper is to present some very easy constructions of S(2, 3, V)‘S and S(2, 4, V)‘S which yield Steiner systems containing three and four prescribed subsystems respectively, and possibly admitting a partition into those subsystems. Furthermore, the subsystems of the obtained new Steiner systems are only the given ones and their possible subsystems. We assume the reader is familiar with the Steiner system terminology and refer him or her to [l, 81 for background and to [6-71 for literature on the subject. A well-known theorem of Doyen [5] states that, given three STS’s 4, i = 1, 2, 3, of the same order u sharing precisely one, possibly empty, subsystem of order w, it is possible to construct an STS of order 3(v w) + w whose only subsystems are the Si’s. Here we consider two naive special cases of Doyen’s construction for two motivations. First, one of these constructions, namely Construction 2, can always be used to produce an S(2, 3, 3(2, w) + w) when the three starting STS(v)‘s share precisely one subsystem of order w 2 0. Secondly, a suitable combination of the two constructions yields a construction of an S(2, 4, 4(v w) + w) by starting with four S(2, 4, V)‘S sharing exactly one, possibly empty, subsystem of order w under the condition v w s 1 (mod 2). In particular, when the four given systems share precisely one point, we obtain an S(2, 4, 4(v 1) + 1) containing the given systems of order v provided that v = 4 (mod 12). Construction 1. This construction of Steiner triple systems works only when the three starting STS(v)‘s are mutually disjoint. Therefore, the obtained STS(3u) admits a partition into three subsystems. 0012-365X/91/$03.50