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A polynomial time construction of binary codes with the currently best known tradeoff between rate and error-correction radius is given. Specifically, linear codes over fixed alphabets are constructed that can be list decoded in polynomial time up to the so-called Blokh-Zyablov bound. The work builds upon earlier work by the authors where codes list decodable up to the Zyablov bound (the standard product bound on distance of concatenated codes) were constructed. The new 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 ldquolevel-by-levelrdquo decoding algorithm, which crucially uses the list recovery algorithm for the outer folded Reed-Solomon codes, enables list decoding up to the designed distance bound, aka the Blokh-Zyablov bound, for multilevel concatenated codes.