On the List-Decodability of Random Linear Codes

For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $ε> 0$, we prove that with high probability a random subspace $C$ of $\F_q^n$ of dimension $(1-H_q(p)-ε)n$ has the property that every Hamming ball of radius $pn$ has at most $O(1/ε)$ codewords. This answers a basic open question concerning the list-decodability of linear codes, showing that a list size of $O(1/ε)$ suffices to have rate within $ε$ 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 $q^{O(1/ε)}$. 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 $Θ(\ell)$) vectors from their linear span also belong to the ball.

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