The Random Coding Bound Is Tight for the Average Linear Code or Lattice

TL;DR

This paper proves that the random-coding bound is tight for average linear codes and lattices, confirming a longstanding conjecture and extending the result to Poltyrev's exponent.

Contribution

It establishes the tightness of the random-coding bound for linear codes and lattices, confirming the conjecture and extending the result to Poltyrev's exponent.

Findings

The random-coding bound is exponentially tight for average linear codes.

The property extends to Poltyrev's random-coding exponent for lattices.

Confirms the conjecture about the behavior of linear code ensembles.

Abstract

In 1973, Gallager proved that the random-coding bound is exponentially tight for the random code ensemble at all rates, even below expurgation. This result explained that the random-coding exponent does not achieve the expurgation exponent due to the properties of the random ensemble, irrespective of the utilized bounding technique. It has been conjectured that this same behavior holds true for a random ensemble of linear codes. This conjecture is proved in this paper. Additionally, it is shown that this property extends to Poltyrev's random-coding exponent for a random ensemble of lattices.