The existence of augmented resolvable Steiner quadruple systems

An augmented Steiner quadruple system of order vv is an ordered triple (X,B,E)(X,B,E), where (X,B)(X,B) is an SQS(v)(v) and EE is the set of all 2-subsets of XX. An augmented Steiner quadruple system (X,B,E)(X,B,E) of order vv is resolvable if B∪E can be partitioned into n=(v−1)(v+4)/6n=(v−1)(v+4)/6 parts B∪E=P1|P2|⋯|PnB∪E=P1|P2|⋯|Pn such that each part PiPi is a partition of XX.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 vv for any positive integer v≡2v≡2 or 10 (mod 12). In this paper, we show that the Hartman and Phelps conjecture is true.