Generators and irreducible polynomials over finite fields
Mathematics of Computation
Decoding of Reed Solomon codes beyond the error-correction bound
Journal of Complexity
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
Complexity of decoding positive-rate primitive Reed-Solomon codes
IEEE Transactions on Information Theory
Counting subset sums of finite abelian groups
Journal of Combinatorial Theory Series A
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
On the subset sum problem over finite fields
Finite Fields and Their Applications
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Under polynomial time reduction, the maximum likelihood decoding of a linear code is equivalent to computing the error distance of a received word. It is known that the decoding complexity of standard Reed-Solomon codes at certain radius is at least as hard as the discrete logarithm problem over certain large finite fields. This implies that computing the error distance is hard for standard Reed-Solomon codes. Using the Weil bound and a new sieve for distinct coordinates counting, we are able to compute the error distance for a large class of received words. This significantly improves previous results in this direction. As a corollary, we also improve the existing results on the Cheng-Murray conjecture about the complete classification of deep holes for standard Reed-Solomon codes.