List decoding algorithms for certain concatenated codes
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
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FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
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FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
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Bounds on list decoding of MDS codes
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Unconditional proof of tightness of Johnson bound
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Explicit capacity-achieving list-decodable codes
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Algorithmic results in list decoding
Foundations and Trends® in Theoretical Computer Science
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AAECC'03 Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes
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We consider the problem of the best possible relation between the list decodability of a binary linear code and its minimum distance. We prove, under a widely-believed number-theoretic conjecture, that the classical "Johnson bound" gives, in general, the best possible relation between the list decoding radius of a code and its minimum distance. The analogous result is known to hold by a folklore random coding argument for the case of non-linear codes, but the linear case is more subtle and has remained open.We prove our result by exhibiting an infinite family of binary linear codes of "large" minimum distance with a super-polynomial number (in blocklength) of codewords all within a Hamming ball of radius close to the Johnson bound. Even the existence of codes with a super-polynomial number of codewords in a ball of radius bounded away from the minimum distance (let alone radius close to the Johnson bound) was open prior to our work. We also unconditionally prove the "tightness" of the Johnson bound for decoding with list size that is an arbitrarily large constant.