  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.
