The existence of augmented resolvable Steiner quadruple systems

Abstract: An augmented Steiner quadruple system of order is an ordered triple , where is an SQS and is the set of all 2-subsets of . An augmented Steiner quadruple system of order is resolvable if can be partitioned into parts such that each part is a partition of . Hartman and Phelps in [A. Hartman, K.T. Phelps, Steiner quadruple systems, in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory, Wiley, New York, 1992, pp. 205–240] conjectured that there exists a resolvable augmented Steiner quadruple systems of order for any positive integer or 10 (mod 12). In this paper, we show that the Hartman and Phelps conjecture is true. [Copyright &y& Elsevier]