Polynomial method in coding and information theory

TL;DR

This paper presents a unified polynomial method framework in coding and information theory, leading to improved asymptotic bounds for codes in various channels and extending previous results.

Contribution

It introduces a general framework for the polynomial method, encompassing prior results and enabling new bounds in coding theory and information theory.

Findings

New asymptotic bounds for codes in binary and Gaussian channels

Extension of polynomial method to a broader class of problems

Improved bounds over classical results from the 1959-67 period

Abstract

Polynomial, or Delsarte's, method in coding theory accounts for a variety of structural results on, and bounds on the size of, extremal configurations (codes and designs) in various metric spaces. In recent works of the authors the applicability of the method was extended to cover a wider range of problems in coding and information theory. In this paper we present a general framework for the method which includes previous results as particular cases. We explain how this generalization leads to new asymptotic bounds on the performance of codes in binary-input memoryless channels and the Gaussian channel, which improve the results of Shannon et al. of 1959-67, and to a number of other results in combinatorial coding theory.