Product constructions for cyclic block designs II. Steiner 2-designs

In [4] we introduced a product construction for cyclic Steiner quadruple systems. The purpose of this paper is to modify the construction so that it is applicable to cyclic Steiner 2-designs. A cyclic Steiner 2-design is a Steiner system s(t, k, u), 2 d t < k < u, in which t = 2 and which has an automorphism of order u. We denote such a system by CS(2, k, u). Under suitable conditions we show how a CS(2, k, uu) may be obtained from a CS(2, k, u) and a CS(2, k, u). CS(2, 3, V) exist for all admissible v (i.e., u = 1 or 3 (mod 6)) apart from u = 9 [6]. However, for k > 3, the values of u for which a CS(2, k, u) exists have not been precisely determined. A survey of known results in this field is given by Colbourn and Mathon in [3]. In [2] Colbourn and Colbourn give product constructions for cyclic balanced incomplete block designs (which are quite different from the constructions presented here). Their constructions, together with ours, considerably extend the spectrum of v for which cyclic systems are known to exist. We believe that the smallest new system generated by our constructions is when k = 5, this system being a CS(2, 5,441) obtained as the product of two CS(2, 5,21)‘s. Given a positive integer u we denote by [u] the set (0, 1,2,..., u 1) and we represent all CS(2, k, v) as unions of k-block orbits formed from elements of [u] under the action of the cyclic group (z + z + 1 (mod u)). The orbits will be referred to as full or (l/n)th orbits according as their 179 0097-3 165/86 $3.00