Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T12:04:06.289Z Has data issue: false hasContentIssue false
  • English
  • Français

Blocking Sets in SQS(2v)

Published online by Cambridge University Press:  12 September 2008

Mario Gionfriddo ,
Salvatore Milici  and
Zsolt Tuza
Mario Gionfriddo
Affiliation:
Dipartimento di Matematica, Città Universitaria, Viale A, Doria 6, 95125 Catania, Italy.
Salvatore Milici
Affiliation:
Dipartimento di Matematica, Città Universitaria, Viale A, Doria 6, 95125 Catania, Italy.
Zsolt Tuza
Affiliation:
Computer and Automation Institute, Hungarian Academy of Sciences, H-llll Budapest, Kende u. 13–17, Hungary
Get access Rights & Permissions [Opens in a new window]

Abstract

A Steiner quadruple system SQS(v) of order v is a family ℬ of 4-element subsets of a v-element set V such that each 3-element subset of V is contained in precisely one B ∈ ℬ. We prove that if T ∩ B ≠ ø for all B ∈ ℬ (i.e., if T is a transversal), then |T| ≥ v/2, and if T is a transversal of cardinality exactly v/2, then V \ T is a transversal as well (i.e., T is a blocking set). Also, in respect of the so-called ‘doubling construction’ that produces SQS(2v) from two copies of SQS(v), we give a necessary and sufficient condition for this operation to yield a Steiner quadruple system with blocking sets.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 1 , March 1994 , pp. 77 - 86
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Berardi, L. and Beutelspacher, A. (to appear) On blocking sets in some block designs.Google Scholar
[2]Doyen, J. and Vandensavel, M. (1971) Non-isomorphic Steiner quadruple systems. Bull. Soc. Math. Belg. 23 393–410.Google Scholar
[3]Eugeni, F. and Mayer, E. (1988) On blocking sets of index two. Annals of Discrete Math. 37 169–176.CrossRefGoogle Scholar
[4]Gionfriddo, M. and Micale, B. (1989) Blocking sets in 3-designs. J. of Geometry, 35 75–86.CrossRefGoogle Scholar
[5]Hanani, H. (1960) On quadruple systems, Canad. J. Math. 12 145–157.CrossRefGoogle Scholar
[6]Phelps, K. T. and Rosa, A. (1980) 2-chromatic Steiner quadruple systems. European J. Comb. 1 253–258.CrossRefGoogle Scholar
[7]Tallini, G. (1983) Blocking sets nei sistemi di Steiner e d-blocking sets in PG(r,q). Quaderno n. 3 Sem. Geom. Combinatorie Univ. L'Aquila.Google Scholar
[8]Tallini, G. (1988) On blocking sets in finite projective and affine spaces. Annals of Discrete Math. 37 433–450.CrossRefGoogle Scholar