Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
On the List and Bounded Distance Decodibility of the Reed-Solomon Codes (Extended Abstract)
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Maximum-likelihood decoding of Reed-Solomon codes is NP-hard
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Improved decoding of Reed-Solomon and algebraic-geometry codes
IEEE Transactions on Information Theory
Explicit theorems on generator polynomials
Finite Fields and Their Applications
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
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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For generalized Reed-Solomon codes, it has been proved [7] that the problem of determining if a received word is a deep hole is co-NP-complete. The reduction relies on the fact that the evaluation set of the code can be exponential in the length of the code - a property that practical codes do not usually possess. In this paper, we first present a much simpler proof of the same result. We then consider the problem for standard Reed-Solomon codes, i.e. the evaluation set consists of all the nonzero elements in the field. We reduce the problem of identifying deep holes to deciding whether an absolutely irreducible hypersurface over a finite field contains a rational point whose coordinates are pairwise distinct and nonzero. By applying Cafure-Matera estimation of rational points on algebraic varieties, we prove that the received vector (f(α))α∈Fpfor the Reed-Solomon [p - 1, k]p, k p1/4-Ɛ, cannot be a deep hole, whenever f(x) is a polynomial of degree k + d for 1 ≤ d ≤ p3/13-Ɛ.