Achieving list decoding capacity using folded reed-solomon codes

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 " > 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-") of errors. At least theoretically, this meets one of the central challenges in algorithmic coding theory. Our codes are simple to describe: they are folded Reed-Solomon codes, which are in fact exactly Reed-Solomon 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 phased bursts. These results were first reported in [1]. The description in this paper, though, is different and the codes are based on a different, more flexible, version of folding. The algebraic argument underlying the decoding algorithm is also simpler, and leads to a slightly better bound on decoding complexity and worst-case list size.