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Multiplicity assignments for algebraic soft-decoding of reed-solomon codes

F. Parvaresh, A. Vardy
2003 IEEE International Symposium on Information Theory, 2003. Proceedings.  
This leads to coding gains of about 0.20dB for RS codes of length 255 and up to 0.75dB for RS codes of length 15, as compared to the Koetter-Vardy algorithm.  ...  This algorithm converts probabilities observed at the channel output into algebraic interpolation conditions, specified in terms of a multiplicity matrix M.  ...  GEOMETRICAL FRAMEWORK Alexander Vardy Department of Electrical Engineering University of California San Diego vardyQkilimanjaro.ucsd.edu We prove that the set of all points M E V with a given cost C (  ... 
doi:10.1109/isit.2003.1228219 fatcat:mhbt7jes7bfqrbfj5jfjaph444

Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421)

Manindra Agrawal, Thomas Thierauf, Christopher Umans, Marc Herbstritt
2013 Dagstuhl Reports  
Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques.  ...  These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting.  ...  condensers based on Parvaresh-Vardy codes.  ... 
doi:10.4230/dagrep.2.10.60 dblp:journals/dagstuhl-reports/AgrawalTU12 fatcat:dg7ithf6xfgadkzwkxjtjhy7ge

Explicit capacity-achieving list-decodable codes

Venkatesan Guruswami, Atri Rudra
2006 Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06  
Further, the decoding of the folded RS codes can be reduced to list decoding the related Parvaresh-Vardy codes. The alphabet size of these folded RS codes is polynomial in the block length.  ...  The main insight in our work is that some carefully chosen folded RS codes are "compressed" versions of a related family of Parvaresh-Vardy codes.  ...  Acknowledgments We thank Charanjit Jutla, Farzad Parvaresh, Anindya Patthak, Madhu Sudan, and Alex Vardy for discussions and comments.  ... 
doi:10.1145/1132516.1132518 dblp:conf/stoc/GuruswamiR06 fatcat:ueq2p5rcnzgldplxrqordnkloa

Performance Analysis of Algebraic Soft-Decision Decoding of Reed-Solomon Codes [article]

Andrew Duggan, Alexander Barg
2007 arXiv   pre-print
These results are also extended to multivariable interpolation in the sense of Parvaresh and Vardy.  ...  We investigate the decoding region for Algebraic Soft-Decision Decoding (ASD) of Reed-Solomon codes in a discrete, memoryless, additive-noise channel.  ...  In this section we extend our analysis of ASD to multivariate interpolation. A code C in the Parvaresh-Vardy (PV) family is defined as follows.  ... 
arXiv:cs/0701018v1 fatcat:urrby7lojjgmflfelfvupfcioe

Approximate common divisors via lattices [article]

Henry Cohn, Nadia Heninger
2012 arXiv   pre-print
In particular, it specializes to a lattice-based list decoding algorithm for Parvaresh-Vardy and Guruswami-Rudra codes, which are multivariate extensions of Reed-Solomon codes.  ...  We analyze the multivariate generalization of Howgrave-Graham's algorithm for the approximate common divisor problem.  ...  Acknowledgements We thank Chris Umans and Alex Vardy for suggesting looking at Parvaresh-Vardy  ... 
arXiv:1108.2714v2 fatcat:u4dmeg7ulbhcljzauzuxtkddj4

Explicit Codes Achieving List Decoding Capacity: Error-correction with Optimal Redundancy [article]

Venkatesan Guruswami, Atri Rudra
2007 arXiv   pre-print
The alphabet size of these folded RS codes is polynomial in the block length.  ...  radius achieved by the standard GMD decoding of concatenated codes.  ...  We thank Piotr Indyk, Charanjit Jutla, Farzad Parvaresh, Anindya Patthak, Madhu Sudan, and Alexander Vardy for useful discussions and comments.  ... 
arXiv:cs/0511072v2 fatcat:o4lzakacxfet3eskhnbhrjhrsi

Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy

Venkatesan Guruswami, Atri Rudra
2008 IEEE Transactions on Information Theory  
The alphabet size of these folded RS codes is polynomial in the block length.  ...  radius achieved by the standard GMD decoding of concatenated codes.  ...  We thank an anonymous referee for several useful comments and in particular for encouraging us to highlight the relevance of the results even for small values of the folding parameter; this was the impetus  ... 
doi:10.1109/tit.2007.911222 fatcat:lsdqe2ohcbbnbavan3t5oarjta

Correlated Algebraic-Geometric Codes: Improved List Decoding over Bounded Alphabets

Venkatesan Guruswami, Anindya Patthak
2006 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)  
Our codes extend the class of algebraic-geometric (AG) codes via a (nonobvious) generalization of the approach in the recent breakthrough work of Parvaresh and Vardy (2005) .  ...  The work of Parvaresh and Vardy (2005) was extended in Guruswami and Rudra (2006) to give explicit codes that achieve the list decoding capacity (optimal trade-off between rate and fraction of errors corrected  ...  Acknowledgments We thank Farzad Parvaresh and Alexander Vardy for sending us an early draft of their paper.  ... 
doi:10.1109/focs.2006.23 dblp:conf/focs/GuruswamiP06 fatcat:zp2t2dtbhvc25b5nmuym73vvfe

Correlated algebraic-geometric codes: Improved list decoding over bounded alphabets

Venkatesan Guruswami, Anindya C. Patthak
2008 Mathematics of Computation  
Our codes extend the class of algebraic-geometric (AG) codes via a (nonobvious) generalization of the approach in the recent breakthrough work of Parvaresh and Vardy (2005) .  ...  The work of Parvaresh and Vardy (2005) was extended in Guruswami and Rudra (2006) to give explicit codes that achieve the list decoding capacity (optimal trade-off between rate and fraction of errors corrected  ...  Acknowledgments We thank Farzad Parvaresh and Alexander Vardy for sending us an early draft of their paper.  ... 
doi:10.1090/s0025-5718-07-02012-1 fatcat:u42tit55x5aydgeigvayf7sudq

Error correction up to the information-theoretic limit

Venkatesan Guruswami, Atri Rudra
2009 Communications of the ACM  
Ever since the birth of coding theory almost 60 years ago, researchers have been pursuing the elusive goal of constructing the "best codes," whose encoding introduces the minimum possible redundancy for  ...  To correct a proportion p (say 20%) of worst-case errors, these codes only need close to a proportion p of redundant symbols. The redundancy cannot possibly be any lower informationtheoretically.  ...  We thank Ronitt Rubinfeld for several valuable comments on an earlier draft of this paper.  ... 
doi:10.1145/1467247.1467269 fatcat:ltwo45d7zjbm5aus3a7qjk4xwe

Folded Algebraic Geometric Codes From Galois Extensions [article]

Ming-Deh Huang, Anand Kumar Narayanan
2009 arXiv   pre-print
This work is an extension of Folded Reed Solomon codes to the setting of Algebraic Geometric codes.  ...  These codes are obtained as a result of folding the set of rational places of a function field using certain elements (automorphisms) from the Galois group of the extension.  ...  Guruswami and Pathak [9] provide a generalization of the Parvaresh-Vardy code to the Algebraic-Geometric setting thereby reducing the alphabet size.  ... 
arXiv:0901.1162v1 fatcat:4457m6w5ljfdlaahkacikihwsa

Cyclotomic function fields, Artin–Frobenius automorphisms, and list error correction with optimal rate

Venkatesan Guruswami
2010 Algebra & Number Theory  
Algebraic error-correcting codes that achieve the optimal trade-off between rate and fraction of errors corrected (in the model of list decoding) were recently constructed by a careful "folding" of the  ...  Using this approach, we construct new folded algebraic-geometric codes for list decoding based on cyclotomic function fields with a cyclic Galois group.  ...  I thank the IAS for providing an inspiring work environment and its wonderful hospitality. Algebra & Number Theory  ... 
doi:10.2140/ant.2010.4.433 fatcat:7y2jl3kkb5dhljfvrasopjfrta

Optimal Rate List Decoding via Derivative Codes [chapter]

Venkatesan Guruswami, Carol Wang
2011 Lecture Notes in Computer Science  
The classical family of [n, k] q Reed-Solomon codes over a field F q consist of the evaluations of polynomials f ∈ F q [X] of degree < k at n distinct field elements.  ...  The algorithm for derivative codes offers some advantages compared to a similar one for folded Reed-Solomon codes in terms of efficient unique decoding in the presence of side information.  ...  Interestingly, it falls in the framework of Parvaresh-Vardy codes, where the correlated polynomials are formal derivatives.  ... 
doi:10.1007/978-3-642-22935-0_50 fatcat:ifpqmfaggbaaphyfmuwoeagjfy

Optimal rate list decoding via derivative codes [article]

Venkatesan Guruswami, Carol Wang
2011 arXiv   pre-print
The classical family of [n,k]_q Reed-Solomon codes over a field _q consist of the evaluations of polynomials f ∈_q[X] of degree < k at n distinct field elements.  ...  The algorithm for derivative codes offers some advantages compared to a similar one for folded Reed-Solomon codes in terms of efficient unique decoding in the presence of side information.  ...  Interestingly, it falls in the framework of Parvaresh-Vardy codes, where the correlated polynomials are formal derivatives.  ... 
arXiv:1106.3951v1 fatcat:d5v5g46435bkjllryoazkrgw4a

Bridging Shannon and Hamming: List Error-correction with Optimal Rate

Venkatesan Guruswami
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
Error-correcting codes tackle the fundamental problem of recovering from errors during data communication and storage. A basic issue in coding theory concerns the modeling of the channel noise.  ...  In particular, we discuss recent explicit constructions of list-decodable codes with information-theoretically optimal redundancy that is arbitrarily close to the fraction of symbols that can be corrupted  ...  Algebraic-geometric codes.  ... 
doi:10.1142/9789814324359_0162 fatcat:ss7ezul2jjemhnxhnl3jjzhwzq
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