Posted by: atri | October 2, 2007

## Lecture 14: List Decoding Capacity

In first part of today’s lecture we saw the proof that $1-H_q(\rho)$ is the list decoding capacity for $q$-ary codes that can list decoded up to $\rho$ fraction of errors.
In the class I mentioned that there exists linear codes with rate $1-H_q(\rho)-\epsilon$ that are $\left(\rho, q^{O(1/\epsilon)}\right)$-list decodable (and this results holds with high probability). I also mentioned that just for $q=2$, one can show the existence of $\left(\rho,O(1/\epsilon)\right)$-list decodable codes. This result (among other things) was proven in the paper titled Combinatorial Bounds for List Decoding by Guruswami, Håstad, Sudan and Zuckerman.