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Design systems: Combinatorial characterizations of Delsarte \(I\)-designs via partially ordered sets. (English) Zbl 0977.05142

Barg, Alexander (ed.) et al., Codes and association schemes. DIMACS workshop, DIMACS Center, Princeton, NJ, USA, November 9-12, 1999. Providence, RI: AMS, American Mathematical Society. DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 56, 223-239 (2001).
The author introduces the notion of a design system. It consists of an association scheme with a partial order on its eigenspaces together with a second partially ordered set having the vertices of the scheme as its maximal elements. He proves, given a design system, that there is an equivalence between certain families of Delsarte \(I\)-designs and designs in the attached partially ordered set. He applies this theory to cometric and Hamming schemes and to the association scheme of the symmetric group. Moreover, he extends two well-known bounds of Delsarte for cometric association schemes [see P. Delsarte, An algebraic approach to the association schemes of coding theory (Philips Res. Reports Suppl. No. 10) (1973)] to the setting of association schemes with many vanishing Krein parameters.
For the entire collection see [Zbl 0960.00079].

MSC:

05E30 Association schemes, strongly regular graphs
05B30 Other designs, configurations
06A11 Algebraic aspects of posets
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