Limits to list decoding Reed-Solomon codes
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Improvements on the Johnson bound for Reed-Solomon codes
Discrete Applied Mathematics
On deciding deep holes of Reed-Solomon codes
TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
Subspace polynomials and limits to list decoding of Reed-Solomon codes
IEEE Transactions on Information Theory
On error correction in the exponent
TCC'06 Proceedings of the Third conference on Theory of Cryptography
Parameter choices on Guruswami--Sudan algorithm for polynomial reconstruction
Finite Fields and Their Applications
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For an error-correcting code and a distance bound, the list decoding problem is to compute all the codewords within a given distance to a received message. The bounded distance decoding problem is to find one codeword if there is at least one codeword within the given distance, or to out-put the empty set if there is not. Obviously the bounded distance decoding problem is not as hard as the list decoding problem. For a Reed-Solomon code [n, k]q, a simple counting argument shows that for any integer 0 0. We show that the discrete logarithm problem over f_q h can be efficiently reduced by a randomized algorithm to the bounded distance decoding problem of the Reed-Solomon code [q, g - h]_q with radius q - g. These results show that the decoding problems for the Reed-Solomon code are at least as hard as the discrete logarithm problem over finite fields. The main tools to obtain these results are an interesting connection between the problem of list-decoding of Reed-Solomon code and the problem of discrete logarithm over finite fields, and a generalization of Katz's theorem on representations of elements in an extension finite field by products of distinct linear factors.