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Euclidean designs and coherent configurations. (English) Zbl 1231.05019
Brualdi, Richard A. (ed.) et al., Combinatorics and graphs. Selected papers based on the presentations at the 20th anniversary conference of IPM on combinatorics, Tehran, Iran, May 15–21, 2009. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4865-4/pbk). Contemporary Mathematics 531, 59-93 (2010).
Summary: The concept of spherical \(t\)-design, which is a finite subset of the unit sphere, was introduced by P. Delsarte, J. M. Goethals and J. J. Seidel [Geom. Dedicata 6, 363–388 (1977; Zbl 0376.05015)] (1977). The concept of Euclidean \(t\)-design, which is a two-step generalization of spherical design in the sense that it is a finite weighted subset of Euclidean space, by A. Neumaier and J. J. Seidel [Indag. Math. 50, No.3, 321-334 (198)].
We first review these two concepts, as well as the concept of tight \(t\)-design, i.e., the one whose cardinality reaches the natural lower bound. We are interested in \(t\)-designs (spherical or Euclidean) which are either tight or close to tight. As is well- known by P. Delsarte, J. M. Goethals and J. J. Seidel [loc. cit.], in the study of spherical \(t\)-designs and in particular of those which are either tight or close to tight, association schemes play important roles.
The main purpose of this paper is to show that in the study of Euclidean \(t\)-designs and in particular of those which are either tight or close to tight, coherent configurations play important roles. Here. coherent configuration is a purely combinatorial concept defined by D. G. Higman [Rend. Sem. Mat. Univ. Padova 44(1970), 1–25 (1971; Zbl 0279.05025) and Geom. Dedicata 4, 1–32 (1975; Zbl 0333.05010)], and is obtained by axiomatizing the properties of general, not necessarily transitive, permutation groups, in the same way as association scheme was obtained by axiomatizing the properties of transitive permutation groups.
In this paper we prove that Euclidean \(t\)-designs satisfying certain conditions give the structure of coherent configurations. Moreover we study the classification problems of Euclidean 4-designs on two concentric spheres with certain additional conditions.
For the entire collection see [Zbl 1202.05003].

MSC:
05B05 Combinatorial aspects of block designs
05E30 Association schemes, strongly regular graphs
51M04 Elementary problems in Euclidean geometries
65D32 Numerical quadrature and cubature formulas
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