On Euclidean designs and the potential energy

TL;DR

This paper explores Euclidean designs through potential energy, establishing bounds using harmonic analysis and linear programming, and introduces strong Euclidean designs with related inequalities and bounds on code cardinalities.

Contribution

It formulates a linear programming bound for potential energy of Euclidean designs and introduces the concept of strong Euclidean designs with Fisher type inequalities.

Findings

Linear programming bounds for potential energy are established.

A new concept of strong Euclidean designs is introduced.

Bounds on the size of Euclidean a-codes and t-designs are derived.

Abstract

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.