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Irreducible designs from supplementary difference sets

Published online by Cambridge University Press:  17 April 2009

D.R. Breach  and
Anne Penfold Street
D.R. Breach
Affiliation:
Department of Mathematics, University of Canterbury, Christchurch, New Zealand
Anne Penfold Street
Affiliation:
Department of Mathematics, University of Queensland, St Lucia, Queensland 4067, Australia.
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Abstract

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A family of n k-subsets of the integers modulo ν are said to be supplementary difference sets if developing them by addition modulo ν leads to a balanced incomplete block design, and to be minimal if no proper subfamily leads to a balanced incomplete block design when developed modulo ν. In other words, the family of supplementary difference sets is minimal precisely when it leads to a balanced incomplete block design which cannot be partitioned into a union of proper subdesigns, each consisting of complete cyclic sets of ν blocks. We discuss the conditions under which such a balanced incomplete block design can be partitioned in some non-cyclic fashion into a union of proper subdesigns.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 31 , Issue 1 , February 1985 , pp. 105 - 115
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Breach, D.R. and Thompson, A.R., “Reducible 2–(11, 5, 14) and 3–(12, 6, 14) designs”, J. Austral. Math. Soc. Ser. A (to appear).Google Scholar
[2]Morgan, Elizabeth J., “Some small quasi-multiple designs”, Ars Combin. 3 (1977), 233–250.Google Scholar
[3]Penfold Street, Anne, “On quasi-multiple designs”, Combinatorial Mathematics V, 206–208 (Lecture Notes in Mathematics, 622. Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar