On spherical codes with inner products in a prescribed interval

TL;DR

This paper introduces a new framework for deriving linear programming bounds on spherical codes with inner products in a specific interval, linking code size bounds to energy bounds, and extending Levenshtein bounds.

Contribution

It develops a unified approach to bounds on spherical codes with prescribed inner product intervals, extending classical Levenshtein bounds and connecting code size and energy bounds.

Findings

New extension of Levenshtein bounds for codes with inner products in [ℓ,s]

Unified derivation valid for various codes and potential functions

Reveals relationship between code cardinality bounds and energy bounds

Abstract

We develop a framework for obtaining linear programming bounds for spherical codes whose inner products belong to a prescribed subinterval $[\ell,s]$ of $[-1,1)$. An intricate relationship between Levenshtein-type upper bounds on cardinality of codes with inner products in $[\ell,s]$ and lower bounds on the potential energy (for absolutely monotone interactions) for codes with inner products in $[\ell,1)$ (when the cardinality of the code is kept fixed) is revealed and explained. Thereby, we obtain a new extension of Levenshtein bounds for such codes. The universality of our bounds is exhibited by a unified derivation and their validity for a wide range of codes and potential functions.