TR10-003 Authors: Venkatesan Guruswami, Johan Håstad, Swastik Kopparty

Publication: 6th January 2010 05:30

Downloads: 3797

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For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $\varepsilon >

0$, we prove that with high probability a random subspace $C$ of

$\F_q^n$ of dimension $(1-H_q(p)-\varepsilon)n$ has the

property that every Hamming ball of radius $pn$ has at most

$O(1/\varepsilon)$ codewords.

This answers a basic open question concerning the list-decodability

of linear codes, showing that a list size of $O(1/\varepsilon)$

suffices to have rate within $\varepsilon$ of the ``capacity''

$1-H_q(p)$. This 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 $q^{O(1/\varepsilon)}$.

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.