Noncommutative Spherical Codes

TL;DR

This paper introduces noncommutative spherical codes within operator algebras, extending classical bounds like the Delsarte-Goethals-Seidel limit to a noncommutative setting, opening new research directions.

Contribution

It generalizes classical spherical code bounds to noncommutative frameworks using Hilbert C*-modules, providing new tools for geometry and coding theory.

Findings

Extension of Pfender's proof to Hilbert C*-modules

Development of noncommutative analogues of classical bounds

New avenues for research in noncommutative geometry and coding

Abstract

Spherical codes, with a rich history spanning nearly five centuries, remain an area of active mathematical exploration and are far from being fully understood. These codes, which arise naturally in problems of geometry, combinatorics, and information theory, continue to challenge researchers with their intricate structure and unresolved questions. Inspired by Polya's heuristic principle of "vary the problem," we extend the classical framework by introducing the notion of noncommutative spherical codes, with particular emphasis on the noncommutative Newton-Gregory kissing number problem. This generalization moves beyond the traditional Euclidean setting into the realm of operator algebras and Hilbert C*-modules, thereby opening new avenues of investigation. A cornerstone in the study of spherical codes is the celebrated Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein linear programming bound, developed over the past half-century. This bound employs Gegenbauer polynomials to establish sharp upper limits on the size of spherical codes, and it has served as a fundamental tool in coding theory and discrete geometry. Remarkably, a recent elegant one-line proof by Pfender [\textit{J. Combin. Theory Ser. A, 2007}] provides a streamlined derivation of a variant of this bound. We demonstrate that Pfender's argument can be extended naturally to the setting of Hilbert C*-modules, thereby enriching the theory with noncommutative analogues.