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 number of nonisomorphic steiner quadruple systems of orders 8, 10, 14, and 16 are 1, 1, 4 (Mendelsohn and Hung 1972), and at least 31,021 (Lindner and Rosa 1976).

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.

Lindner, C. L. and Rosa, A. There are at Least 31,021 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.