Advertisement

Journal of Statistical Theory and Practice

, Volume 6, Issue 1, pp 97–128 | Cite as

Steiner Quadruple Systems With Point-Regular Abelian Automorphism Groups

Article

Abstract

In this article we present a graph theoretic construction of Steiner quadruple systems (SQS) admitting Abelian groups as point-regular automorphism groups. The resulting SQS has an extra property that we call A-reversibility, where A is the underlying Abelian group. In particular, when A is a 2-group of exponent at most 4, it is shown that an A-reversible SQS always exists. When the Sylow 2-subgroup of A is cyclic, we give a necessary and sufficient condition for the existence of an A-reversible SQS, which is a generalization of a necessary and sufficient condition for the existence of a dihedral SQS by Piotrowski (1985). This enables one to construct A-reversible SQS for any Abelian group A of order v such that for every prime divisor p of v there exists a dihedral SQS(2p).

05E20 05B05 

Keywords

Combinatorial design Graph Finite group Steiner system 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bitan, S., and T. Etzion. 1993. The last packing number of quadruples, and cyclic SQS. Des. Codes Cryptogr., 3, 283–313.MathSciNetCrossRefGoogle Scholar
  2. Dembowski, P. 1968. Finite geometries. Springer.CrossRefGoogle Scholar
  3. Feng, T, Y. Chang, and L. Ji. 2008. Constructions for strictly cyclic 3-designs and applications to optimal OOCs with λ = 2. J. Combin. Theory Ser. A, 115, 1527–1551.MathSciNetCrossRefGoogle Scholar
  4. Fitting, F. 1915. Zyklische Lösung des Steinerschen Problems. Nieuw. Arch. Wisk., 11, 140–148.MATHGoogle Scholar
  5. Hanani, H. 1963. On some tactical configurations. Can. J. Math., 15, 705–722.MathSciNetCrossRefGoogle Scholar
  6. Huber, M. 2010. Almost simple groups with socle Ln(q) acting on Steiner quadruple systems. J. Combin. Theory Ser. A, 117, 1004–1007.MathSciNetCrossRefGoogle Scholar
  7. Kaski, P., P. R. J. Östergård, and O. Pottonen. 2006. The Steiner quadruple systems of order 16. J. Combin. Theory Ser. A, 113, 1764–1770.MathSciNetCrossRefGoogle Scholar
  8. Köhler, E. 1979. Zyklische Quadrupelsysteme. Abh. Math. Sem. Univ. Hamburg, 48, 1–24.MathSciNetCrossRefGoogle Scholar
  9. Lovász, L. 1993. Combinatorial problems and exercises (2nd ed.). North-Holland, Amsterdam.MATHGoogle Scholar
  10. Munemasa, A., and M. Sawa. 2007. Simple abelian quadruple systems. J. Combin. Theory Ser. A, 114, 1160–1164.MathSciNetCrossRefGoogle Scholar
  11. Petersen, J. 1891. Die Theorie der regulären graphs. Acta Math., 15, 193–220.MathSciNetCrossRefGoogle Scholar
  12. Piotrowski, W. 1985. Untersuchungen über S-zyklische Quadrupelsysteme. Dissertation, University of Hamburg.MATHGoogle Scholar
  13. Sawa, M. 2010. Optical orthogonal signature pattern codes of weight 4 and maximum collision parameter 2. IEEE Trans. Inform. Theory, 56, 3613–3620.MathSciNetCrossRefGoogle Scholar
  14. Siemon, H. 1987. Some remarks on the construction of cyclic Steiner quadruple systems. Arch. Math. (Basel), 49, 166–178.MathSciNetCrossRefGoogle Scholar
  15. Siemon, H. 1991. On the existence of cyclic Steiner quadruple systems SQS(2p). Discrete Math., 97, 377–385.MathSciNetCrossRefGoogle Scholar
  16. Siemon, H. 1998. A number-theoretic conjecture and the existence of S-cyclic Steiner quadruple systems. Des. Codes Cryptogr., 13, 63–94.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
  2. 2.Graduate School of Information SciencesNagoya UniversityNagoyaJapan

Personalised recommendations