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 Steiner System 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 a Steiner triple system of order (Ryser 1963, p. 101).
Examples of Steiner triple systems of small orders are
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 the
finite Projective Plane of order 2. is a finite Affine Plane which can be constructed from the array
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
A030129
in ``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.
© 1996-9 Eric W. Weisstein