Hostname: page-component-cc8bf7c57-j4qg9 Total loading time: 0 Render date: 2024-12-11T05:22:44.487Z Has data issue: false hasContentIssue false
  • English
  • Français

On the minimum number of blocks defining a design

Published online by Cambridge University Press:  17 April 2009

Ken Gray
Ken Gray
Affiliation:
Department of Mathematics, University of Queensland, St Lucia Qld 4067, Australia
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.

A set of blocks which is a subset of a unique t – (v, k, λt) design is said to be a defining set of that design. We examine the properties of such a set, and show that its automorphism group is related to that of the whole design. Smallest defining sets are found for 2-designs and 3-designs on seven or eight varieties with block size three or four, revealing interesting combinatorial structures.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 41 , Issue 1 , February 1990 , pp. 97 - 112
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Biggs, N.L. and White, A.T., Permutation Groups and Combinatorial Structures (Cambridge University Press, Cambridge, 1979).CrossRefGoogle Scholar
[2]Billington, Elizabeth J., ‘Further Constructions of Irreducible Designs’, Congressus Numerantium 35 (1982), 77–89.Google Scholar
[3]Breach, D.R., ‘The 2 – (9,4,3) and 3 – (10,5,3) Designs’, Combin. Theory, Ser. A 27 (1979), 50–63.CrossRefGoogle Scholar
[4]Cannon, John, A Language for Group Theory (Department of Pure Mathematics, University of Sydney, 1987).Google Scholar
[5]Gray, Ken, ‘Designs Carried by a Code’, Ars Combin. 23B (1987), 257–271.Google Scholar
[6]Hughes, D.R. and Piper, F.C., Design Theory (Cambridge University Press, Cambridge, 1985).CrossRefGoogle Scholar
[7]McKay, Brendan D., nauty User's Guide (Version 1.2) (Australian National University Computer Science Technical Report TR-CS-87-03, 1987).Google Scholar
[8]Morgan, Elizabeth J., ‘Some small quasi-multiple designs’, Ars Combin. 3 (1977), 233–250.Google Scholar
[9]Nandi, Harikinkar, ‘A further note on non-isomorphic solutions of incomplete block designs’, Sankhya 7 (1945-1946), 313–316.Google Scholar
[10]Rodger, C.A., ‘Triple Systems with a Fixed Number of Repeated Blocks’, J. Austral. Math. Soc. (A) 41 (1986), 180–187.CrossRefGoogle Scholar
[11]Stanton, R.G. and Collens, R.J., ‘A computer system for research on the family classification of BIBDs’, Proc.Internat.Congress on Combinatorial Theory, Accad. dei Lincei (Rome, 1973), I (1976), 133–169.Google Scholar
[12]Street, Anne Penfold and Street, Deborah J., Combinatorics of Experimental Design (Clarendon Press, Oxford, 1987).Google Scholar
[13]Wallis, W.D., private communication; see Morgan [8].Google Scholar