On Euclidean designs and potential energy (Q426741)

Summary: We study Euclidean designs from the viewpoint of the potential energy. For a finite set in Euclidean space, we formulate a linear programming bound for the potential energy by applying harmonic analysis on a sphere. We also introduce the concept of strong Euclidean designs from the viewpoint of the linear programming bound, and we give a Fisher type inequality for strong Euclidean designs. A finite set on Euclidean space is called a Euclidean \(a\)-code if any distinct two points in the set are separated at least by \(a\). As a corollary of the linear programming bound, we give a method to determine an upper bound on the cardinalities of Euclidean \(a\)-codes on concentric spheres of given radii. Similarly we also give a method to determine a lower bound on the cardinalities of Euclidean \(t\)-designs as an analogue of the linear programming bound.