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).
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. A113, 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."