On the ListDecodability of Random Linear Codes
Abstract
For every fixed finite field $\F_q$, $p \in (0,11/q)$ and $\epsilon > 0$, we prove that with high probability a random subspace $C$ of $\F_q^n$ of dimension $(1H_q(p)\epsilon)n$ 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 listdecodability of linear codes, showing that a list size of $O(1/\epsilon)$ suffices to have rate within $\epsilon$ of the "capacity" $1H_q(p)$. Our result matches up to constant factors the listsize achieved by general random codes, and gives an exponential improvement over the best previously known listsize 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2010
 arXiv:
 arXiv:1001.1386
 Bibcode:
 2010arXiv1001.1386G
 Keywords:

 Computer Science  Information Theory;
 Mathematics  Combinatorics
 EPrint:
 15 pages