Expander-Based Constructions of Efficiently Decodable Codes
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
List Decoding of Error-Correcting Codes: Winning Thesis of the 2002 ACM Doctoral Dissertation Competition (Lecture Notes in Computer Science)
Explicit capacity-achieving list-decodable codes
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
List decoding and property testing of error-correcting codes
List decoding and property testing of error-correcting codes
Concatenated codes can achieve list-decoding capacity
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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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 [7] 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 [7], enables list decoding up to the designed distance bound, aka the Blokh-Zyablov bound, for multilevel concatenated codes.