Abstract:
Various code constructions use expander graphs to improve the error resilience. Often the use of expanding graphs comes at the expense of the alphabet size. This is the case, e.g., in \cite{abnnr}, \cite{guruswami01expanderbased} and \cite{Guru04}. We show that by replacing the balanced expanding graphs used in the above constructions with unbalanced dispersers or extractors depending on the actual construction) the alphabet size can be dramatically improved.

Abstract:
In TR04-069 we revisit the construction of high noise, almost optimal rate list decodable codes of Guruswami \cite{Guru04}, and note that by replacing the balanced expander component in his construction with an unbalanced disperser the alphabet size can be dramatically improved. In this report we point out that the same type of improvement can be done in various other expander based constructions given by Guruswami and Indyk in \cite{guruswami01expanderbased}. We give here one example. \cite{guruswami01expanderbased} give an explicit encodable/decodable unique decoding with rate $\epsilon$, relative distance $(1-\epsilon)$ and alphabet size $2^{O({1 \over \epsilon})}$. we improve the alphabet size to $2^{2^{polyloglog({1 \over \epsilon})}}$ (assuming one can build an explicit optimal extractor the alphabet size can get to $poly({1 \over \epsilon})$). Unlike the construction in \cite{Guru04}, where the expanding graph is used only for its expansion properties, the construction studied here requires both expansion and mixing. Thus, the improvement is achieved by replacing the balanced expander used with an unbalanced extractor.

Abstract:
In this note we revisit the construction of high noise, almost optimal rate list decodable code of Guruswami ("Better extractors for better codes?") Guruswami showed that based on optimal extractors one can build a $(1-\epsilon,O({1 \over \epsilon}))$ list decodable codes of rate $\Omega({\epsilon \over {log{1 \over \epsilon}}})$ and alphabet size $2^{O({1 \over \epsilon} \cdot {log{1 \over \epsilon}})}$. We show that if one replaces the expander component in the construction with an unbalanced disperser, than one can improve the alphabet size to $2^{O(log^2{1 \over \epsilon})}$ while keeping all other parameters the same.