Design systems: Combinatorial characterizations of Delsarte \(I\)-designs via partially ordered sets (Q2717208)

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 \textit{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.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00079].