Linear Programming Hierarchies in Coding Theory: Dual Solutions

TL;DR

This paper develops the first dual feasible solutions for a hierarchy of linear programs in coding theory, potentially leading to improved bounds on the rate versus distance problem for linear codes.

Contribution

It introduces the first dual solutions for the LP hierarchy, matching best-known bounds and paving the way for better upper bounds in coding theory.

Findings

First dual feasible solutions matching best-known bounds

Potential for improved upper bounds on rate vs. distance

Advances in linear programming approaches for coding theory

Abstract

The rate vs. distance problem is a long-standing open problem in coding theory. Recent papers have suggested a new way to tackle this problem by appealing to a new hierarchy of linear programs. If one can find good dual solutions to these LPs, this would result in improved upper bounds for the rate vs. distance problem of linear codes. In this work, we develop the first dual feasible solutions to the LPs in this hierarchy. These match the best-known bound for a wide range of parameters. Our hope is that this is a first step towards better solutions, and improved upper bounds for the rate vs. distance problem of linear codes.