## 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 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

One of the two s is a finite Hyperbolic Plane. The 80 Steiner triple systems have been studied by Tonchev and Weishaar (1997). There are more than Steiner triple systems of order 19 (Stinson and Ferch 1985; Colbourn and Dinitz 1996, p. 15).

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.