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Maximum-likelihood decoding of Reed-Solomon Codes is NP-hard [article]

Venkatesan Guruswami, Alexander Vardy
2004 arXiv   pre-print
In this paper, we prove that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes.  ...  Maximum-likelihood decoding is one of the central algorithmic problems in coding theory. It has been known for over 25 years that maximum-likelihood decoding of general linear codes is NP-hard.  ...  It is particularly interesting that the only nontrivial family of codes for which we can now prove that maximum-likelihood decoding is NP-hard is the family of Reed-Solomon codes.  ... 
arXiv:cs/0405005v1 fatcat:dtt2nptws5codkb2vthb6w7syu

Maximum-Likelihood Decoding of Reed–Solomon Codes is NP-Hard

V. Guruswami, A. Vardy
2005 IEEE Transactions on Information Theory  
In this paper, we prove that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes.  ...  In this paper, we prove that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes. Let q © 2 m and let £ q X denote the ring of univariate polynomials over £ q .  ...  As discussed in Corollary 9, maximum-likelihood decoding of Reed-Solomon codes becomes hard when the number of errors is large -one less than the covering radius of the code.  ... 
doi:10.1109/tit.2005.850102 fatcat:huoqs7n4gvebxkbgvmgai7xate

Hard Problems of Algebraic Geometry Codes

Qi Cheng
2008 IEEE Transactions on Information Theory  
The minimum distance is one of the most important combinatorial characterizations of a code. The maximum likelihood decoding problem is one of the most important algorithmic problems of a code.  ...  In general, much less is known about the hardness of the specific classes of natural linear codes. In this paper, we show that both problems are NP-hard for algebraic geometry codes.  ...  The only result of this kind known to date is the result of [7] on the NP-completeness of maximum likelihood decoding for generalized Reed-Solomon codes, where the sizes of the alphabets are exponential  ... 
doi:10.1109/tit.2007.911213 fatcat:wcu25ouofbenpjd2ywsxxizmou

New Set of Codes for the Maximum-Likelihood Decoding Problem [article]

Morgan Barbier
2010 arXiv   pre-print
The maximum-likelihood decoding problem is known to be NP-hard for general linear and Reed-Solomon codes.  ...  For these codes, we show that the maximum-likelihood decoding problem is reachable in polynomial time in the code parameters.  ...  However there may be quite a few of those codes as we exhibited nine binary BCH codes which are Wu-covered codes, of which, four constitute a new result and two can be decoded in time quasi-quadratic in  ... 
arXiv:1011.2834v1 fatcat:opy5a4ipsbagrgdwbiluidfdkm

Hard Problems of Algebraic Geometry Codes [article]

Qi Cheng
2005 arXiv   pre-print
The minimum distance is one of the most important combinatorial characterizations of a code. The maximum likelihood decoding problem is one of the most important algorithmic problems of a code.  ...  In general, much less is known about the hardness of the specific classes of natural linear codes. In this paper, we show that both problems are NP-hard for algebraic geometry codes.  ...  The only result of this kind to date is the result of [10] on the NP-completeness of maximum likelihood decoding for Reed-Solomon codes.  ... 
arXiv:cs/0507026v1 fatcat:txre47ut7rdtvhsxejbb4hynfm

Complexity of Decoding Positive-Rate Reed-Solomon Codes [chapter]

Qi Cheng, Daqing Wan
2008 Lecture Notes in Computer Science  
The complexity of maximum likelihood decoding of the Reed-Solomon codes [q − 1, k]q is a well known open problem.  ...  In particular, this resolves an open problem left in [4] , and rules out the possibility of a polynomial time algorithm for maximum likelihood decoding problem of Reed-Solomon codes of any rate under a  ...  The maximum likelihood decoding of a generalized Reed-Solomon [n, k] q code is known to be NP-complete [6] .  ... 
doi:10.1007/978-3-540-70575-8_24 fatcat:4hjtug54xncffbg5bacrvnyrqu

On Deciding Deep Holes of Reed-Solomon Codes [article]

Qi Cheng, Elizabeth Murray
2005 arXiv   pre-print
For generalized Reed-Solomon codes, it has been proved GuruswamiVa05 that the problem of determining if a received word is a deep hole is co-NP-complete.  ...  We then consider the problem for standard Reed-Solomon codes, i.e. the evaluation set consists of all the nonzero elements in the field.  ...  This shows that maximum likelihood decoding of Reed-Solomon codes, as well as the bounded distance decoding at radius n − k − 1, is at least as hard as deciding deepholes.  ... 
arXiv:cs/0509065v1 fatcat:lxaciygbsncwhizrc4kphm6ogy

Algorithmic complexity in coding theory and the minimum distance problem

Alexander Vardy
1997 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC '97  
We start withan overviewof algorithmiccomplexity problemsin coding theory We then show that the problemof computing the minimumdiktanceof a binaryIinwr code is NP-hard,and the corresponding deci~"on problemis  ...  This constitutes a proof of the conjecture Bedekamp, McEliece,vanTilborg, dating back to 1978. Extensionsand applicationsof this result to other problemsin codingtheqv are discussed.  ...  I am especially indebted to Noga Alon for referring me to his construction, which is used in Section 3. Finally, I would like to thank Hagit Itzkowitz for her invaluable help.  ... 
doi:10.1145/258533.258559 dblp:conf/stoc/Vardy97 fatcat:vkunvltuabfdvnjzldvt6mpihu

Algebraic soft-decision decoding of reed-solomon codes

R. Koetter, A. Vardy
2003 IEEE Transactions on Information Theory  
A polynomial-time soft-decision decoding algorithm for Reed-Solomon codes is developed.  ...  The resulting soft-decoding algorithm significantly outperforms both the Guruswami-Sudan decoding and the generalized minimum distance (GMD) decoding of Reed-Solomon codes, while maintaining a complexity  ...  They thank the anonymous referee for valuable comments that improved the presentation of this paper.  ... 
doi:10.1109/tit.2003.819332 fatcat:5bewpwi4zrfstpg47z6i7avqh4

Algorithms for Modular Counting of Roots of Multivariate Polynomials [chapter]

Parikshit Gopalan, Venkatesan Guruswami, Richard J. Lipton
2006 Lecture Notes in Computer Science  
We show an equivalence between maximum-likelihood decoding for Reed-Solomon codes and a root-finding problem for symmetric polynomials.  ...  We give an efficient algorithm to compute N r (P ) when the modulus r is a power of the characteristic of the field. We show that for all other moduli, the problem of computing N r (P ) is NP-hard.  ...  We show that maximum-likelihood decoding of Reed-Solomon codes is related to a variant of this problem.  ... 
doi:10.1007/11682462_51 fatcat:pdjtzzwmorgipmtiddmyy3kruq

Algorithms for Modular Counting of Roots of Multivariate Polynomials

Parikshit Gopalan, Venkatesan Guruswami, Richard J. Lipton
2007 Algorithmica  
We show an equivalence between maximum-likelihood decoding for Reed-Solomon codes and a root-finding problem for symmetric polynomials.  ...  We give an efficient algorithm to compute N r (P ) when the modulus r is a power of the characteristic of the field. We show that for all other moduli, the problem of computing N r (P ) is NP-hard.  ...  We show that maximum-likelihood decoding of Reed-Solomon codes is related to a variant of this problem.  ... 
doi:10.1007/s00453-007-9097-3 fatcat:75pwlendejetrkjo43ttgguriy

Decoding of Reed Solomon Codes beyond the Error-Correction Bound

Madhu Sudan
1997 Journal of Complexity  
This immediately provides a maximum likelihood decoding algorithm for Reed Solomon Codes, which works in a setting with a larger number of errors than any previously known algorithm.  ...  To the best of our 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  ...  I am especially grateful to Greg Sorkin for spending numerous hours implementing a version of the algorithm presented here.  ... 
doi:10.1006/jcom.1997.0439 fatcat:r5p5sy5rfnhvbeextz2gdyf6im

The intractability of computing the minimum distance of a code

A. Vardy
1997 IEEE Transactions on Information Theory  
It is shown that the problem of computing the minimum distance of a binary linear code is NP-hard, and the corresponding decision problem is NP-complete.  ...  Extensions and applications of this result to other problems in coding theory are discussed.  ...  For example, we could let be the binary subfield subcode of , as is commonly done in obtaining BCH codes from Reed-Solomon codes.  ... 
doi:10.1109/18.641542 fatcat:tn6dlz2fk5cufl6mnfl6zzwqnu

On the threshold of Maximum-Distance Separable codes

Bruno Kindarji, Gerard Cohen, Herve Chabanne
2010 2010 IEEE International Symposium on Information Theory  
Or, dually speaking, is Maximum-Likelihood decoding almost surely impossible?  ...  In a second part, we explicit lower-bounds on the threshold of Maximum-Distance Separable codes such as Reed-Solomon codes, and compute the threshold for the toy example that motivates this study.  ...  This result is coherent with previous results on the hardness of decoding Reed-Solomon codes.  ... 
doi:10.1109/isit.2010.5513670 dblp:conf/isit/KindarjiCC10 fatcat:iep65am72rdsvpyj7ymaerzvve

A Novel Efficient Sphere Decoding with Reed-Solomon Code in MIMO Detection

2019 VOLUME-8 ISSUE-10, AUGUST 2019, REGULAR ISSUE  
In this article, motivated by this idea, we present novel efficient detection-decoding combinational techniques using Reed-Solomon (RS) decoding followed by the single tree search (STS) algorithm in SD  ...  This system can employ a coding strategy for information security where the data symbol space forms a scant lattice. The sparsity structure is resolved through channel code.  ...  ACKNOWLEDGMENT This work has endorsed by the Department of Electronics, under the project funded by Sardar Vallabhbhai National Institute of Technology (SVNIT).  ... 
doi:10.35940/ijitee.i8909.078919 fatcat:ia3l42l7grcmlm7vzv42n6qoq4
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