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).
A cyclic Steiner quadruple system identifies its points with 
so that translations preserve
the blocks. It is symmetric cyclic if negation preserves each translation group
orbit of blocks. Liu and Feng (2026) proved that a symmetric cyclic Steiner quadruple
system of order 
exists iff 
and every odd prime
divisor 
of 
satisfies 
or 
.
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.Liu, H. and Feng, T. "The Existence of Symmetric
Cyclic Steiner Quadruple Systems." 19 Oct 2026. 
and 
." Utilitas Math. 1, 5-95, 1972.Sloane,
N. J. A. Sequence