Energy Bounds for Codes in Polynomial Metric Spaces

TL;DR

This paper develops universal lower bounds on the potential energy of codes in polynomial metric spaces using linear programming, applicable across various spaces like spheres and Hamming spaces, with asymptotic analysis for large dimensions.

Contribution

It introduces universally optimal lower bounds for code energy in polynomial metric spaces, unifying and extending previous bounds across multiple space types.

Findings

Universal lower bounds apply to Euclidean spheres, projective, Hamming, and Johnson spaces.

Asymptotic bounds derived for large code size and dimension.

Connections established between energy bounds and code separation constraints.

Abstract

In this article we present a unified treatment for obtaining bounds on the potential energy of codes in the general context of polynomial metric spaces (PM-spaces). The lower bounds we derive via the linear programming (LP) techniques of Delsarte and Levenshtein are universally optimal in the sense that they apply to a broad class of energy functionals and, in general, cannot be improved for the specific subspace. Tests are presented for determining whether these universal lower bounds (ULB) can be improved on larger spaces. Our ULBs are applicable on the Euclidean sphere, infinite projective spaces, as well as Hamming and Johnson spaces. Asymptotic results for the ULB for the Euclidean spheres and the binary Hamming space are derived for the case when the cardinality and dimension of the space grow large in a related way. Our results emphasize the common features of the Levenshtein's universal upper bounds for the cardinality of codes with given separation and our ULBs for energy. We also introduce upper bounds for the energy of designs in PM-spaces and the energy of codes with given separation.