  The concept of spherical $t$-design, which is a finite subset of the unit
sphere, was introduced by Delsarte-Goethals-Seidel (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
Neumaier-Seidel (1988). 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
Delsarte-Goethals-Seidel (1977), 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, 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. The main purpose of this paper is to 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.
