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On generalised \(t\)-designs and their parameters. (English) Zbl 1226.05056

Summary: Recently, P.J. Cameron studied a class of block designs which generalises the classes of \(t\)-designs, \(\alpha \)-resolved 2-designs, orthogonal arrays, and other classes of combinatorial designs. In fact, Cameron’s generalisation of \(t\)-designs (when there are no repeated blocks) is essentially a special case of the “poset \(t\)-designs” in product association schemes studied ten years earlier by W.J. Martin, who further studied the special case of “mixed block designs”
In this paper, we study Cameron’s generalisation of \(t\)-designs from the point of view of classical \(t\)-design theory, in particular investigating the parameters of these generalised \(t\)-designs. We show that the \(t\)-design constants \(\lambda _i\) (the number of blocks containing an \(i\)-subset of the points, where \(i\leq t\)) and \(\lambda ^j_i\) (the number of blocks containing an \(i\)-subset \(I\) of the points and disjoint from a \(j\)-subset \(J\) of the points, where \(I\cap J=\emptyset \) and \(i+j\leq t\)) have very natural counterparts for generalised \(t\)-designs. Our main result places strong restrictions on the block structure of Cameron’s \(t\)-\((v, k, \lambda )\) designs, an important subclass of generalised \(t\)-designs. We also generalise N.S. Mendelsohn’s concept of “intersection numbers of order \(r\)” for \(t\)-designs, and show that analogous equations to those of Mendelsohn hold for generalised \(t\)-designs.

MSC:

05B05 Combinatorial aspects of block designs

Software:

GAP; DESIGN; Maple
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Full Text: DOI

References:

[1] Cameron, P. J., A generalisation of \(t\)-designs, Discrete Math., 309, 4835-4842 (2009) · Zbl 1186.05024
[2] Cameron, P. J.; Soicher, L. H., Block intersection polynomials, Bull. Lond. Math. Soc., 39, 559-564 (2007) · Zbl 1131.05014
[3] The GAP Group, GAPhttp://www.gap-system.org/; The GAP Group, GAPhttp://www.gap-system.org/
[4] Maple, Version 13, 2009, http://www.maplesoft.com/; Maple, Version 13, 2009, http://www.maplesoft.com/
[5] Martin, W. J., Mixed block designs, J. Combin. Des., 6, 151-163 (1998) · Zbl 0918.05017
[6] Martin, W. J., Designs in product association schemes, Des. Codes Cryptogr., 16, 271-289 (1999) · Zbl 0927.05082
[7] Martin, W. J.; Tanaka, H., Commutative association schemes, European J. Combin., 30, 1497-1525 (2009) · Zbl 1228.05317
[8] Mendelsohn, N. S., Intersection numbers of \(t\)-designs, (Mirsky, L., Studies in Pure Mathematics (1971), Academic Press: Academic Press London), 145-150 · Zbl 0222.05018
[9] Ray-Chaudhuri, D. K.; Wilson, R. M., On \(t\)-designs, Osaka J. Math., 12, 737-744 (1975) · Zbl 0342.05018
[10] L.H. Soicher, The DESIGNGAPhttp://designtheory.org/software/gap_design/; L.H. Soicher, The DESIGNGAPhttp://designtheory.org/software/gap_design/
[11] Soicher, L. H., More on block intersection polynomials and new applications to graphs and block designs, J. Combin. Theory, Ser. A., 117, 799-809 (2010) · Zbl 1228.05080
[12] van Trung, T.; Wu, Q.; Mesner, D. M., High order intersection numbers of \(t\)-designs, J. Statist. Plann. Inference, 56, 257-268 (1996) · Zbl 0873.05014
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