A Steiner quadruple system is a Steiner system 
of integer order 
, consisting of a set 
of 
elements and a collection 
of four-element subsets of 
such that every three-element subset of 
is contained in exactly one
member of 
.
Barrau (1908) established the uniqueness up to isomorphism
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).
Nonisomorphic Steiner Quadruple Systems of Order 16."
Utilitas Math. 10, 61-64, 1976.Lindner, C. C. 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