On error distance of Reed-Solomon codes
@article{Li2008OnED,
title={On error distance of Reed-Solomon codes},
author={Yujuan Li and Daqing Wan},
journal={Science in China Series A: Mathematics},
year={2008},
volume={51},
pages={1982-1988}
}The complexity of decoding the standard Reed-Solomon code is a well known open problem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, we show that the error distance can be determined precisely when the degree of the received word is small. As an application of our method, we give a significant improvement of the recent bound of Cheng-Murray on non-existence of deep holes (words with maximal error distance…
32 Citations
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References
SHOWING 1-8 OF 8 REFERENCES
On Deciding Deep Holes of Reed-Solomon Codes
- Computer ScienceTAMC
- 2007
By applying Cafure-Matera estimation of rational points on algebraic varieties, it is proved that the received vector (f(α))α∈Fp for the Reed-Solomon, cannot be a deep hole, whenever f(x) is a polynomial of degree k + d for 1 ≤ d ≤ p3/13-Ɛ.
Maximum-likelihood decoding of Reed-Solomon codes is NP-hard
- Computer ScienceIEEE Transactions on Information Theory
- 2005
It is proved that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes, thereby strengthening a result of Bruck and Naor.
On the List and Bounded Distance Decodibility of the Reed-Solomon Codes (Extended Abstract)
- Computer ScienceFOCS
- 2004
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.
On the list and bounded distance decodibility of Reed-Solomon codes
- Computer Science45th Annual IEEE Symposium on Foundations of Computer Science
- 2004
This paper proves that list decoding can not be done for radius n - g(n, k: q) or larger, otherwise the discrete logarithm over F/sub qg(m, k, q)-k/ is easy, and shows that the discretelogarithM problem over F-sub qh/ can be efficiently reduced by a randomized algorithm to the bounded distance decoding problem of the Reed-Solomon code.
On the subset sum problem over finite fields
- Computer Science, MathematicsFinite Fields Their Appl.
- 2008
Improved decoding of Reed-Solomon and algebraic-geometric codes
- Computer ScienceProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998
An improved list decoding algorithm for decoding Reed-Solomon codes and alternant codes and algebraic-geometric codes is presented, including a solution to a weighted curve fitting problem, which is of use in soft-decision decoding algorithms for Reed- Solomon codes.
Decoding of Reed Solomon Codes beyond the Error-Correction Bound
- Computer ScienceJ. Complex.
- 1997
To the best of the knowledge, this is the first efficient (i.e., polynomial time bounded) algorithm which provides error recovery capability beyond the error-correction bound of a code for any efficient code.
Generators and irreducible polynomials over finite fields
- MathematicsMath. Comput.
- 1997
An application to the distribution of irreducible polynomials is given, which confirms an asymptotic version of a conjecture of Hansen-Mullen about the construction of generators for the multiplicative group of a finite field.