Subspace Polynomials and List Decoding of Reed-Solomon Codes
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
On the List and Bounded Distance Decodability of Reed-Solomon Codes
SIAM Journal on Computing
On deciding deep holes of Reed-Solomon codes
TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
The number field sieve in the medium prime case
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
Improved decoding of Reed-Solomon and algebraic-geometry codes
IEEE Transactions on Information Theory
Maximum-likelihood decoding of Reed-Solomon codes is NP-hard
IEEE Transactions on Information Theory
Limits to List Decoding Reed–Solomon Codes
IEEE Transactions on Information Theory
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
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It has been proved that the maximum likelihood decoding problem of Reed-Solomon codes is NP-hard. However, the length of the code in the proof is at most polylogarithmic in the size of the alphabet. For the complexity of maximum likelihood decoding of the primitive Reed-Solomon code, whose length is one less than the size of alphabet, the only known result states that it is at least as hard as the discrete logarithm in some cases where the information rate unfortunately goes to zero. In this paper, it is proved under a well known cryptography hardness assumption that: 1) There does not exist a randomized polynomial time maximum likelihood decoder for the Reed-Solomon code family [q, k(q)]q, where k(x) is any function in Z+ → Z+ computable in time xO(1) satisfying √x ≤ k(x) ≤ x - √x. 2) There does not exist a randomized polynomial time bounded-distance decoder for primitive Reed-Solomon codes at distance 2/3 + Ζ of the minimum distance for any constant 0 ≤ Ζ ≤ 1/3. In particular, this rules out the possibility of a polynomial time algorithm for maximum likelihood decoding problem of primitive Reed-Solomon codes of any rate under the assumption.