Steiner Quadruple System
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 theuniqueness of 
,
and
Fitting (1915) subsequently constructed the cyclic systems
and 
, and Bays and de Weck (1935) showed theexistence of at least one 
. Hanani (1960) proved that a Necessary and Sufficient condition forthe 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).
See also Steiner System, Steiner Triple System
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.
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