On the difficulty to beat the first linear programming bound for binary codes

TL;DR

This paper discusses the challenges in surpassing the first linear programming bound for binary codes, highlighting difficulties faced when extending current methods towards the second bound.

Contribution

It analyzes the limitations of recent approaches in extending the linear programming bounds for binary codes beyond the first bound.

Findings

Recent methods face fundamental difficulties beyond the first bound.

Extending bounds requires overcoming specific technical challenges.

Current techniques may not be sufficient for tighter bounds.

Abstract

The first linear programming bound of McEliece, Rodemich, Rumsey, and Welch is the best known asymptotic upper bound for binary codes, for a certain subrange of distances. Starting from the work of Friedman and Tillich, there are, by now, some arguably easier and more direct arguments for this bound. We show that this more recent line of argument runs into certain difficulties if one tries to go beyond this bound (say, towards the second linear programming bound of McEliece, Rodemich, Rumsey, and Welch).