On the List-Decodability of Random Linear Codes

TL;DR

This paper proves that random linear codes near the capacity limit are highly list-decodable with small list sizes, matching the performance of general codes and significantly improving previous bounds.

Contribution

It establishes that random linear codes of rate close to capacity have small list sizes, resolving an open question in the list-decodability of linear codes.

Findings

Random linear codes have list size O(1/ε) near capacity.

Improves previous list size bounds from exponential to polynomial in 1/ε.

Provides a key probabilistic bound on vectors in Hamming balls.

Abstract

For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $\epsilon > 0$, we prove that with high probability a random subspace $C$ of $\F_q^n$ of dimension $(1-H_q(p)-\epsilon)n$ has the property that every Hamming ball of radius $pn$ has at most $O(1/\epsilon)$ codewords. This answers a basic open question concerning the list-decodability of linear codes, showing that a list size of $O(1/\epsilon)$ suffices to have rate within $\epsilon$ of the "capacity" $1-H_q(p)$. Our result matches up to constant factors the list-size achieved by general random codes, and gives an exponential improvement over the best previously known list-size bound of $q^{O(1/\epsilon)}$. The main technical ingredient in our proof is a strong upper bound on the probability that $\ell$ random vectors chosen from a Hamming ball centered at the origin have too many (more than $\Theta(\ell)$) vectors from their linear span also belong to the ball.