Improving the Alphabet-Size in High Noise, Almost Optimal Rate List Decodable Codes

Abstract

We revisit the construction of high noise, almost optimal rate list decodable code of Guruswami [1]. Guruswami showed that if one can explicitly construct optimal extractors then one can build an explicit $(1-\epsilon,O(\frac{1}{\epsilon}))$ list decodable codes of rate $\Omega(\frac{\epsilon}{log \frac{1}{\epsilon}})$ and alphabet size $2^{O(\frac{1}{\epsilon}\cdot log\frac{1}{\epsilon})}$. We show that if one replaces the expander component in the construction with an unbalanced disperser, then one can dramatically improve the alphabet size to $2^{O(log^{2}\frac{1}{\epsilon})}$ while keeping all other parameters the same.