# Maximum-likelihood decoding of Reed-Solomon codes is NP-hard

@article{Guruswami2004MaximumlikelihoodDO, title={Maximum-likelihood decoding of Reed-Solomon codes is NP-hard}, author={Venkatesan Guruswami and Alexander Vardy}, journal={Electron. Colloquium Comput. Complex.}, year={2004} }

Maximum-likelihood decoding is one of the central problems in coding theory. It has been known for over 25 years that maximum-likelihood decoding of general linear codes is NP-hard. Nevertheless, it was so far unknown whether maximum-likelihood decoding remains hard for any specific family of codes with nontrivial algebraic structure. In this paper, we prove that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes. We moreover show that maximum-likelihood decoding of…

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## 45 Citations

Computing Error Distance of Reed-Solomon Codes

- Mathematics, Computer ScienceTAMC
- 2012

Using the Weil bound and a new sieve for distinct coordinates counting, this work is able to compute the error distance for a large class of received words and improves the existing results on the Cheng-Murray conjecture about the complete classification of deep holes for standard Reed-Solomon codes.

Iterative Algebraic Soft Decision Decoding of Reed-Solomon Codes

- Mathematics
- 2004

In this paper, we propose an iterative soft decision decoding scheme for Reed Solomon codes with near maximum likelihood performance. The advantage of this decoding algorithm over previously proposed…

On error distance of received words with fixed degrees to Reed-Solomon code

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- 2015

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…

On Reed-Solomon codes

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- 2011

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…

Efficient soft decoding techniques for reed-solomon codes

- Mathematics
- 2010

The main focus of this thesis is on finding efficient decoding methods for Reed-Solomon (RS) codes, i.e., algorithms with acceptable performance and affordable complexity. Three classes of decoders…

On the error distance of extended Reed-Solomon codes

- Mathematics, Computer ScienceAdv. Math. Commun.
- 2016

Using some algebraic constructions, this work is able to determine the error distance of words whose degrees are $k+1$ and $k-2$ to the extended Reed-Solomon codes.

On the threshold of Maximum-Distance Separable codes

- Computer Science, Mathematics2010 IEEE International Symposium on Information Theory
- 2010

This paper deals with the threshold of linear q-ary error-correcting codes, and shows that when the minimal distance of the code is high enough, the threshold effect is very sharp.

On the NP-hardness of bounded distance decoding of Reed-Solomon codes

- Mathematics, Computer Science2015 IEEE International Symposium on Information Theory (ISIT)
- 2015

This work extends the result of Guruswami and Vardy by proving that it is NP-hard to decide the existence of a codeword that disagrees with v on n - k - 2, and on n- k - 3 coordinates.

Algebraic list-decoding of error-correcting codes

- Computer Science
- 2007

This dissertation develops a new decoding algorithm for Reed-Solomon codes, which decodes some M codewords together and extends the bivariate polynomial interpolation method of Guruswami-Sudan to multivariate interpolation decoding, and develops a sub-optimal solution based upon optimization of second- order statistics.

On deep holes of generalized Reed-Solomon codes

- Mathematics
- 2016

Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word u is a deep hole of the standard Reed-Solomon codes [q-1, k] q…

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