Steiner Triple System
Let 
be a set of 
elements together with a set 
of 3-subset (triples) of such that every 2-Subset of occurs in exactly one triple of . Then is called a Steiner triple system and is a special case of a SteinerSystem with 
and 
. A Steiner triple system 
of order 
exists Iff 
(Kirkman 1847). In addition, if Steiner triple systems 
and 
of orders 
and 
exist, then so does aSteiner triple system 
of order 
(Ryser 1963, p. 101).
Examples of Steiner triple systems 
of small orders are
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The number of nonisomorphic Steiner triple systems of orders 
, 9, 13, 15, 19, ... (i.e., 
) are 1,1, 2, 80, 
, ... (Colbourn and Dinitz 1996, pp. 14-15; Sloane's A030129). 
is the same as thefinite Projective Plane of order 2. 
is a finite Affine Plane which can be constructed from the array
One of the two
s is a finite Hyperbolic Plane. The 80 Steiner triple systems 
have been studied byTonchev and Weishaar (1997). There are more than 
Steiner triple systems of order 19 (Stinson and Ferch1985; Colbourn and Dinitz 1996, p. 15).See also Hadamard Matrix, Kirkman Triple System, Steiner Quadruple System, Steiner System
References
Colbourn, C. J. and Dinitz, J. H. (Eds.) ``Steiner Triple Systems.'' §4.5 in CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, pp.14-15 and 70, 1996. Kirkman, T. P. ``On a Problem in Combinatorics.'' Cambridge Dublin Math. J. 2, 191-204, 1847. Lindner, C. C. and Rodger, C. A. Design Theory. Boca Raton, FL: CRC Press, 1997. Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 99-102, 1963. Sloane, N. J. A. Sequence A030129in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html. Stinson, D. R. and Ferch, H. ``2000000 Steiner Triple Systems of Order 19.'' Math. Comput. 44, 533-535, 1985. Tonchev, V. D. and Weishaar, R. S. ``Steiner Triple Systems of Order 15 and Their Codes.'' J. Stat. Plan. Inference 58, 207-216, 1997.
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