Explicit Codes Achieving List Decoding Capacity: Error-correction with Optimal Redundancy

Abstract

We present error-correcting codes that achieve the information-theoretically best possible trade-off between the rate and error-correction radius. Specifically, for every $0 < R < 1$ and $\eps> 0$, we present an explicit construction of error-correcting codes of rate $R$ that can be list decoded in polynomial time up to a fraction $(1-R-\eps)$ of {\em worst-case} errors. At least theoretically, this meets one of the central challenges in algorithmic coding theory. Our codes are simple to describe: they are {\em folded Reed-Solomon codes}, which are in fact {\em exactly} Reed-Solomon (RS) codes, but viewed as a code over a larger alphabet by careful bundling of codeword symbols. Given the ubiquity of RS codes, this is an appealing feature of our result, and in fact our methods directly yield better decoding algorithms for RS codes when errors occur in {\em phased bursts}. The alphabet size of these folded RS codes is polynomial in the block length. We are able to reduce this to a constant (depending on $\eps$) using ideas concerning ``list recovery'' and expander-based codes from \cite{GI-focs01,GI-ieeejl}. Concatenating the folded RS codes with suitable inner codes also gives us polynomial time constructible binary codes that can be efficiently list decoded up to the Zyablov bound, i.e., up to twice the radius achieved by the standard GMD decoding of concatenated codes.