Thresholds of Random Quasi-Abelian Codes

TL;DR

This paper establishes that the Gilbert-Varshamov bound acts as a threshold for the relative distance of random quasi-abelian codes, showing a phase transition in their properties based on the code rate.

Contribution

It proves the GV-bound is a threshold for the relative distance of random quasi-abelian codes, demonstrating the existence of many asymptotically good codes with parameters reaching the GV-bound.

Findings

GV-bound is a threshold for code distance probability

Existence of many asymptotically good quasi-abelian codes

Phase transition in code properties at GV-bound

Abstract

For a random quasi-abelian code of rate $r$, it is shown that the GV-bound is a threshold point: if $r$ is less than the GV-bound at $\delta$, then the probability of the relative distance of the random code being greater than $\delta$ is almost 1; whereas, if $r$ is bigger than the GV-bound at $\delta$, then the probability is almost 0. As a consequence, there exist many asymptotically good quasi-abelian codes with any parameters attaining the GV-bound.