Abstract
A transformation of Steiner quadruple systems S(υ, 4, 3) is introduced. For a given system, it allows to construct new systems of the same order, which can be nonisomorphic to the given one. The structure of Steiner systems S(υ, 4, 3) is considered. There are two different types of such systems, namely, induced and singular systems. Induced systems of 2-rank r can be constructed by the introduced transformation of Steiner systems of 2-rank r − 1 or less. A sufficient condition for a Steiner system S(υ, 4, 3) to be induced is obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hanani, H., On Quadruple Systems, Canad. J. Math., 1960, vol. 12, pp. 145–157.
Hartman, A., The Existence of Resolvable Steiner Quadruple Systems, J. Combin. Theory, Ser. A, 1987, vol. 44, no. 2, pp. 182–206.
Ji, L. and Zhu, L., Resolvable Steiner Quadruple Systems for the Last 23 Orders, SIAM J. Discrete Math., 2005, vol. 19, no. 2, pp. 420–430.
Lindner, C.C. and Rosa, A., Steiner Quadruple Systems—A Survey, Discrete Math., 1978, vol. 22, no. 2, pp. 147–181.
Hartman, A. and Phelps, K.T., Steiner Quadruple Systems, Contemporary Design Theory: A Collection of Surveys, Ch. 6, Dinitz, J.H. and Stinson, D.R., Eds., New York: Wiley, 1992, pp. 205–240.
The CRC Handbook of Combinatorial Designs, Colbourn, C.J. and Dinitz, J.H., Eds., Boca Raton: CRC Press, 1996.
Tonchev, V.D., A Formula for the Number of Steiner Quadruple Systems on 2n Points of 2-Rank 2n−n, J. Combin. Des., 2003, vol. 11, no. 4, pp. 260–274.
Zinoviev, V.A. and Zinoviev, D.V., Classification of Steiner Quadruple Systems of Order 16 and Rank at Most 13, Probl. Peredachi Inf., 2004, vol. 40, no. 4, pp. 48–67 [Probl. Inf. Trans. (Engl. Transl.), 2004, vol. 40, no. 4, pp. 337–355].
Zinoviev, V.A. and Zinoviev, D.V., Classification of Steiner Quadruple Systems of Order 16 and Rank 14, Probl. Peredachi Inf., 2006, vol. 42, no. 3, pp. 59–72 [Probl. Inf. Trans. (Engl. Transl.), 2006, vol. 42, no. 3, pp. 217–229].
Kaski, P., Östergård, P.R.J., and Pottonen, O., The Steiner Quadruple Systems of Order 16, J. Combin. Theory, Ser. A, 2006, vol. 113, no. 8, pp. 1764–1770.
Zinoviev, V.A. and Zinoviev, D.V., On Resolvability of Steiner Systems S(υ = 2m, 4, 3) of Rank r ≤ υ − m + 1 over \( \mathbb{F}_2 \), Probl. Peredachi Inf., 2007, vol. 43, no. 1, pp. 39–55 [Probl. Inf. Trans. (Engl. Transl.), 2007, vol. 43, no. 1, pp. 33–47].
Semakov, N.V. and Zinoviev, V.A., Constant-Weight Codes and Tactical Configurations, Probl. Peredachi Inf., 1969, vol. 5, no. 3, pp. 29–38 [Probl. Inf. Trans. (Engl. Transl.), 1969, vol. 5, no. 3, pp. 22–28].
Doyen, J., Hubaut, X., and Vandensavel, M., Ranks of Incidence Matrices of Steiner Triple Systems, Math. Z., 1978, vol. 163, no. 3, pp. 251–259.
Doyen, J. and Vandensavel, M., Nonisomorphic Steiner Quadruple Systems, Bull. Soc. Math. Belg., 1971, vol. 23, pp. 393–410.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.A. Zinoviev, D.V. Zinoviev, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 4, pp. 26–42.
Supported in part by the Russian Foundation for Basic Research, project no. 09-01-00536.
Rights and permissions
About this article
Cite this article
Zinoviev, V.A., Zinoviev, D.V. On one transformation of Steiner quadruple systems S(υ, 4, 3). Probl Inf Transm 45, 317–332 (2009). https://doi.org/10.1134/S0032946009040036
Received:
Accepted:
Published:
Issue date:
DOI: https://doi.org/10.1134/S0032946009040036