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Steiner Quadruple System

A Steiner quadruple system is a Steiner system S(t=3,k=4,v), where S is a v-set and B is a collection of k-sets of S such that every t-subset of S is contained in exactly one member of B. Barrau (1908) established the uniqueness of S(3,4,8),

1 2 4 8; 2 3 5 8; 3 4 6 8; 4 5 7 8; 1 5 6 8; 2 6 7 8; 1 3 7 8 3 5 6 7; 1 4 6 7; 1 2 5 7; 1 2 3 6; 2 3 4 7; 1 3 4 5; 2 4 5 6

and S(3,4,10)

1 2 4 5; 2 3 5 6; 3 4 6 7; 4 5 7 8; 5 6 8 9; 6 7 9 0; 1 7 8 0; 1 2 8 9; 2 3 9 0; 1 3 4 0 1 2 3 7; 2 3 4 8; 3 4 5 9; 4 5 6 0; 1 5 6 7; 2 6 7 8; 3 7 8 9; 4 8 9 0; 1 5 9 0; 1 2 6 0 1 3 5 8; 2 4 6 9; 3 5 7 0; 1 4 6 8; 2 5 7 9; 3 6 8 0; 1 4 7 9; 2 5 8 0; 1 3 6 9; 2 4 7 0.

Fitting (1915) subsequently constructed the cyclic systems S(3,4,26) and S(3,4,34), and Bays and de Weck (1935) showed the existence of at least one S(3,4,14). Hanani (1960) proved that a necessary and sufficient condition for the existence of an S(3,4,v) is that v=2 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).

See also

Steiner System, Steiner Triple System

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References

Barrau, J. A. "On the Combinatory Problem of Steiner." K. Akad. Wet. Amsterdam Proc. Sect. Sci. 11, 352-360, 1908.Bays, S. and de Weck, E. "Sur les systèmes de quadruples." Comment. Math. Helv. 7, 222-241, 1935.Fitting, F. "Zyklische Lösungen des Steiner'schen Problems." Nieuw Arch. Wisk. 11, 140-148, 1915.Hanani, M. "On Quadruple Systems." Canad. J. Math. 12, 145-157, 1960.Kaski, P.; Östergård, P. R. J.; and Pottonen, O. "The Steiner Quadruple Systems of Order 16." J. Combin. Th. Ser. A 113, 1764-1770, 2006.Lindner, C. L. and Rosa, A. "There are at Least 31021 Nonisomorphic Steiner Quadruple Systems of Order 16." Utilitas Math. 10, 61-64, 1976.Lindner, C. L. 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 S(3,4,14) and S(4,5,15)." Utilitas Math. 1, 5-95, 1972.Sloane, N. J. A. Sequence A124119 in "The On-Line Encyclopedia of Integer Sequences."

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Steiner Quadruple System

Cite this as:

Weisstein, Eric W. "Steiner Quadruple System." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteinerQuadrupleSystem.html

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