A functional view of upper bounds on codes

TL;DR

This paper explores the mathematical foundations of upper bounds on codes using functional and linear-algebraic methods, linking orthogonal polynomials, kernels, and eigenfunctions to optimize code bounds.

Contribution

It introduces a new perspective by connecting Christoffel-Darboux kernels and Levenshtein polynomials to eigenfunctions of the Jacobi operator, enhancing the theoretical framework for code bounds.

Findings

Christoffel-Darboux kernels are stationary points of moment functionals

Levenshtein polynomials arise as eigenfunctions of the Jacobi operator

Provides a unified view of polynomial-based bounds in coding theory

Abstract

Functional and linear-algebraic approaches to the Delsarte problem of upper bounds on codes are discussed. We show that Christoffel-Darboux kernels and Levenshtein polynomials related to them arise as stationary points of the moment functionals of some distributions. We also show that they can be derived as eigenfunctions of the Jacobi operator. This motivates the choice of polynomials used to derive linear programming upper bounds on codes in homogeneous spaces.