Hostname: page-component-54dcc4c588-42vt5 Total loading time: 0 Render date: 2025-10-13T08:35:13.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Pair-packings and projective planes

Part of: Designs and configurations

Published online by Cambridge University Press:  09 April 2009

D. R. Stinson
D. R. Stinson
Affiliation:
Department of Computer Science University of ManitobaWinnipeg, Manitoba R3T 2N2, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An (n + 1, n2 + n + 1)-packing is a collection of blocks, each of size n + 1, chosen from a set of size n2 + n + 1, such that no pair of points is contained in more than one block. If any two blocks contain a common point, then the packing can be extended to a projective plane of order n, provided the number of blocks is sufficiently large. We study packings which have a pair of disjoint blocks (such a packing clearly cannot be extended to a projective plane of order n). No such packing can contain more than n2 + n/2 blocks. Also, if n is the order of a projective plane, then we can construct such a packing with n2 + 1 blocks.

MSC classification

Secondary: 05B25: Finite geometries 05B40: Packing and covering

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 37 , Issue 1 , August 1984 , pp. 27 - 38
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Baker, R. D., ‘An elliptic semiplane’, J. Combinatorial Theory Ser. A 25 (1978), 193–195.CrossRefGoogle Scholar
[2]Dow, S., ‘An improved bound for extending partial projective planes’, Discrete Math., to appear.Google Scholar
[3]Rosa, A. and Kotzig, A., ‘Nearly Kirkman systems’, Proc. Fifth Southeastern Conf. on Combinatorics, Graph Theory and Computing, Boca Raton, La., 1974, pp. 607–614.Google Scholar
[4]Schellenberg, P. J., ‘Further results on Vp (7,1),’ Proc. Eighth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Baton Rouge, La., 1977, pp. 591–604.Google Scholar
[5]Stinson, D. R., ‘The non-existence of certain finite linear spaces,’ Geom. Dedicata, to appear.Google Scholar
[6]Wallis, W. D., Street, A. P. and Wallis, J. S., Combinatorics: Room squares, sum-free sets, Hadamard metrics, (Lecture Notes in Math., vol. 292, Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar