Limits to list decoding Reed-Solomon codes
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Tensor-based hardness of the shortest vector problem to within almost polynomial factors
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Algorithmic results in list decoding
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List decoding tensor products and interleaved codes
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A deterministic reduction for the gap minimum distance problem: [extended abstract]
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An efficient algorithm to find all small-size stopping sets of low-density parity-check matrices
IEEE Transactions on Information Theory
Provably good codes for hash function design
SAC'06 Proceedings of the 13th international conference on Selected areas in cryptography
On the hardness of approximating stopping and trapping sets
IEEE Transactions on Information Theory
On the De Boer-Pellikaan method for computing minimum distance
Journal of Symbolic Computation
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List Decoding Tensor Products and Interleaved Codes
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FSTTCS'06 Proceedings of the 26th international conference on Foundations of Software Technology and Theoretical Computer Science
On bounded distance decoding for general lattices
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We show that the minimum distance d of a linear code is not approximable to within any constant factor in random polynomial time (RP), unless nondeterministic polynomial time (NP) equals RP. We also show that the minimum distance is not approximable to within an additive error that is linear in the block length n of the code. Under the stronger assumption that NP is not contained in random quasi-polynomial time (RQP), we show that the minimum distance is not approximable to within the factor 2log1-ε(n), for any ε0. Our results hold for codes over any finite field, including binary codes. In the process, we show that it is hard to find approximately nearest codewords even if the number of errors exceeds the unique decoding radius d/2 by only an arbitrarily small fraction εd. We also prove the hardness of the nearest codeword problem for asymptotically good codes, provided the number of errors exceeds (2/3)d. Our results for the minimum distance problem strengthen (though using stronger assumptions) a previous result of Vardy (1997) who showed that the minimum distance cannot be computed exactly in deterministic polynomial time (P), unless P = NP. Our results are obtained by adapting proofs of analogous results for integer lattices due to Ajtai (1998) and Micciancio (see SIAM J. Computing, vol.30, no.6, p.2008-2035, 2001). A critical component in the adaptation is our use of linear codes that perform better than random (linear) codes.