Abstract
This paper deals with the following question: how many, and which, blocks of a design with given parameters must be known before the remaining blocks of the design are uniquely determined? We survey the theoretical background on such defining sets, some specific results for smallest and other minimal defining sets for small designs and the techniques used in finding them, the few known results on minimal defining sets for infinite classes of designs, and the conjectures on minimal defining sets for some classes of Hadamard designs.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Peter J Cameron, How few entries determine any Latin square? Bulletin of the Institute of Combinatorics and its Applications 10 (1994), 63–65.
John Cannon, A language for group theory, Department of Pure Mathematics, University of Sydney, Sydney, Australia (1987).
R T Curtis, Eight octads suffice, Journal of Combinatorial Theory, Series A 36 (1984), 116–123.
Cathy Delaney, private communication, (1994).
Rebecca A H Gower, Minimal defining sets in a family of Steiner triple systems, Australasian Journal of Combinatorics 8 (1993), 55–73.
Rebecca A H Gower, Minimal defining sets for another infinite family of Steiner triple systems, submitted for publication.
Ken Gray, On the minimum number of blocks defining a design, Bulletin of the Australian Mathematical Society 41 (1990), 97–112.
Ken Gray, Further results on smallest defining sets of well known designs, Australasian Journal of Combinatorics 1 (1990), 91–100.
Ken Gray, Defining sets of single-transposition-free designs, Utilitas Math- ematica 38 (1990), 97–103.
Ken Gray and Anne Penfold Street, Smallest defining sets of the five non- isomorphic 2—(15,7,3) designs, Bulletin of the Institute of Combinatorics and its Applications 9 (1993), 96–102.
Ken Gray and Anne Penfold Street, The smallest defining set of the 2- (15,7,3) designs associated with PG(3,2): a theoretical approach, Bulletin of the Institute of Combinatorics and its Applications 11 (1994), 77–83.
Catherine S Greenhill, An algorithm for finding smallest defining sets of t-designs, Journal of Combinatorial Mathematics and Combinatorial Computing 14 (1993), 39–60.
Catherine S Greenhill and Anne Penfold Street, Smallest defining sets of some small t-designs and relations to the Petersen graph, Utilitas Mathematica, to appear.
A Khodkar, Smallest defining sets for the 36 non-isomorphic twofold triple systems of order nine, Journal of Combinatorial Mathematics and Combinatorial Computing, to appear.
Thomas Kunkle and Jennifer Seberry, A few more small defining sets for SBIBD(4t–1,2t– 1,t - 1), submitted for publication.
Brendan D McKay, nauty User’s Guide (Version 1.5), Australian National University Computer Science Technical Report TR-CS-90–02.
Tony Moran, Smallest defining sets of 2—(9,4,3) and 3—(10,5,3) designs, Australasian Journal of Combinatorics 10 (1994), 265–288.
Tony Moran, Defining sets of the 2-(19,9,4) designs and a class of Hadamard designs,submitted for publication.
Tony Moran, private communication, (1993).
Dinesh Sarvate and. Jennifer Seberry, A note on small defining sets for some SBIBD(4t—1, 2t—1, t—1), Bulletin of the Institute of Combinatorics and its Applications 10 (1994), 26–32.
Jennifer Seberry, On small defining sets for some SBIBD(4t — 1,21 — 1,2 — 1), Bulletin of the Institute of Combinatorics and its Applications 4 (1992), 58–62; corrigendum, 6 (1992), 62.
Martin J Sharry and Sylvia H Elmes, The program COMPLETE, private communication, (1992).
Anne Penfold Street, Defining sets for t-designs and critical sets for Latin squares, New Zealand Journal of Mathematics 21 (1992), 133–144.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Kluwer Academic Publishers. Printed in the Netherlands
About this chapter
Cite this chapter
Street, A.P. (1995). Defining Sets for Block Designs: An Update. In: Colbourn, C.J., Mahmoodian, E.S. (eds) Combinatorics Advances. Mathematics and Its Applications, vol 329. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3554-2_22
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3554-2_22
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3556-6
Online ISBN: 978-1-4613-3554-2
eBook Packages: Springer Book Archive