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A Steiner quadruple system is a Steiner system , where is a -set and is a collection of -sets of such that every -subset of is contained in exactly one member of . Barrau (1908) established the uniqueness of ,

and

Fitting (1915) subsequently constructed the cyclic systems and , and Bays and de Weck (1935) showed the existence of at least one . Hanani (1960) proved that a necessary and sufficient condition for the existence of an is that or 4 (mod 6).

The numbers of nonisomorphic Steiner quadruple systems of orders 8, 10, 14, 16, ... are 1, 1, 4 (Mendelsohn and Hung 1972), 1054163 (Kaski et al. 2006), ... (OEIS A124119).

Steiner System, Steiner Triple System

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## References

Barrau, J. A. "On the Combinatory Problem of Steiner." K. Akad. Wet. Amsterdam Proc. Sect. Sci. 11, 352-360, 1908.Bays, S. and de Weck, E. "Sur les systèmes de quadruples." Comment. Math. Helv. 7, 222-241, 1935.Fitting, F. "Zyklische Lösungen des Steiner'schen Problems." Nieuw Arch. Wisk. 11, 140-148, 1915.Hanani, M. "On Quadruple Systems." Canad. J. Math. 12, 145-157, 1960.Kaski, P.; Östergård, P. R. J.; and Pottonen, O. "The Steiner Quadruple Systems of Order 16." J. Combin. Th. Ser. A 113, 1764-1770, 2006.Lindner, C. L. and Rosa, A. "There are at Least Nonisomorphic Steiner Quadruple Systems of Order 16." Utilitas Math. 10, 61-64, 1976.Lindner, C. L. and Rosa, A. "Steiner Quadruple Systems--A Survey." Disc. Math. 22, 147-181, 1978.Mendelsohn, N. S. and Hung, S. H. Y. "On the Steiner Systems and ." Utilitas Math. 1, 5-95, 1972.Sloane, N. J. A. Sequence A124119 in "The On-Line Encyclopedia of Integer Sequences."