Abstract:
We give a polynomial time construction of binary codes with the best currently known trade-off between rate and error-correction radius. Specifically, we obtain linear codes over fixed alphabets that can be list decoded in polynomial time up to the so called Blokh-Zyablov bound. Our work builds upon (Guruswami-Rudra STOC06) where codes list decodable up to the Zyablov bound (the standard product bound on distance of concatenated codes) were constructed. Our codes are constructed via a (known) generalization of code concatenation called multilevel code concatenation. A probabilistic argument, which is also derandomized via conditional expectations, is used to show the existence of inner codes with a certain nested list decodability property that is appropriate for use in multilevel concatenated codes. A ``level-by-level'' decoding algorithm, which crucially uses the list recovery algorithm for folded Reed-Solomon codes from (Guruswami-Rudra STOC06), enables list decoding up to the designed distance bound, aka the Blokh-Zyablov bound, for multilevel concatenated codes.