A deterministic reduction for the gap minimum distance problem: [extended abstract]
Proceedings of the forty-first annual ACM symposium on Theory of computing
Complexity of decoding positive-rate primitive Reed-Solomon codes
IEEE Transactions on Information Theory
Hardness of Reconstructing Multivariate Polynomials over Finite Fields
SIAM Journal on Computing
On the subset sum problem over finite fields
Finite Fields and Their Applications
Computing error distance of reed-solomon codes
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
Generalised jacobians in cryptography and coding theory
WAIFI'12 Proceedings of the 4th international conference on Arithmetic of Finite Fields
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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 output 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 ${\bf F}_{q^{h}}$ 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 certain finite fields. For the list decoding problem of Reed-Solomon codes, although the infeasible radius that we obtain is much larger than the radius, which is known to be feasible, it is the first nontrivial bound. Our result on the bounded distance decodability of Reed-Solomon codes is also the first of its kind. The main tools for obtaining these results are an interesting connection between the problem of list decoding of Reed-Solomon code, the problem of a 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.