Functional Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Bound

TL;DR

This paper introduces a new nonlinear upper bound for spherical codes and the kissing number problem by extending classical bounds to pointed metric spaces, including Banach spaces, inspired by Pfender's simplified proof.

Contribution

It generalizes the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein bounds to pointed metric spaces and introduces a nonlinear approach to the kissing number problem.

Findings

Derived a nonlinear upper bound for spherical codes.

Extended bounds to Banach spaces and pointed metric spaces.

Proposed a nonlinear formulation of the kissing number problem.

Abstract

Pfender \textit{[J. Combin. Theory Ser. A, 2007]} provided a one-line proof for a variant of the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein upper bound for spherical codes, which offers an upper bound for the celebrated (Newton-Gregory) kissing number problem. Motivated by this proof, we introduce the notion of codes in pointed metric spaces (in particular on Banach spaces) and derive a nonlinear (functional) Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender upper bound for spherical codes. We also introduce nonlinear (functional) Kissing Number Problem.