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Ideal-theoretic Explanation of Capacity-achieving Decoding [article]

Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, Madhu Sudan
2021 arXiv   pre-print
We show that good bounds on this distance lead to capacity-achieving performance of the underlying code, providing a unifying explanation of known capacity-achieving results.  ...  Specifically, we present a unified view of the decoding algorithm for ideal theoretic codes and show that the decodability reduces to the analysis of the distance of some related codes.  ...  Regarding the aspect of the need to lower bound the distance of the code generated by Diag(L), to bound the list size of the codes, we stress that for each of these codes the lower bound on the distance  ... 
arXiv:2103.07930v1 fatcat:4vzcjnekezgllokqtbiztbbkqa

Power Error Locating Pairs [article]

Alain Couvreur, Isabella Panaccione
2020 arXiv   pre-print
We present a new decoding algorithm based on error locating pairs and correcting an amount of errors exceeding half the minimum distance.  ...  It permits to get an abstract description of decoding using only codes and linear algebra and without involving the arithmetic of polynomial and rational function algebras used for the definition of the  ...  Acknowledgements The authors express their gratitude to the anonymous referees for their careful work and their many relevant comments permitting a significant improvement of this article.  ... 
arXiv:1907.11658v3 fatcat:vealqwmqo5a2zbxcs73pzyw7qy

Subspace Polynomials and List Decoding of Reed-Solomon Codes

Eli Ben-Sasson, Swastik Kopparty, Jaikumar Radhakrishnan
2006 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)  
We show combinatorial limitations on efficient list decoding of Reed-Solomon codes beyond the Johnson and Joh63, GS99] .  ...  In particular, we show that for arbitrarily large fields F N , |F N | = N , for any δ ∈ (0, 1), and K = N δ : * Research supported by an Alon fellowship of the Israeli council for higher education, by  ...  Acknowledgements We thank Prahladh Harsha for several helpful discussions on subspace and linearized polynomials.  ... 
doi:10.1109/focs.2006.73 dblp:conf/focs/Ben-SassonKR06 fatcat:3j5yqhy32bhxrflhgr2rles4ee

Optimal rate list decoding of folded algebraic-geometric codes over constant-sized alphabets [chapter]

Venkatesan Guruswami, Chaoping Xing
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms  
In this paper, we will employ the method used in [8, 6] for list decoding the Reed-Solomon codes and cyclotomic codes to bound the list size.  ...  Optimal rate list decoding of folded algebraic-geometric codes over constant-sized alphabets * (Extended abstract) Venkatesan Guruswami † Chaoping Xing ‡ Abstract We construct a new list-decodable family  ... 
doi:10.1137/1.9781611973402.134 dblp:conf/soda/GuruswamiX14 fatcat:6hzluttv3jashefielsuk3m67u

List-decoding of binary Goppa codes up to the binary Johnson bound [article]

Daniel Augot, Morgan Barbier (INRIA Saclay - Ile de France), Alain Couvreur
2010 arXiv   pre-print
We study the list-decoding problem of alternant codes, with the notable case of classical Goppa codes.  ...  The major consideration here is to take into account the size of the alphabet, which shows great influence on the list-decoding radius.  ...  where δ is the minimum distance of the Generalised Reed-Solomon code from which the alternant code is built.  ... 
arXiv:1012.3439v1 fatcat:wapmpvtxjjetbjyasm5wzmmewm

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

Venkatesan Guruswami
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
We survey recent progress on list decoding, highlighting its power and generality as an avenue to construct codes resilient to worst-case errors with information rates similar to what is possible against  ...  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.  ...  List decoding of Reed-Solomon codes Consider the Reed-Solomon code RS F,S [n, k] from Definition 1.  ... 
doi:10.1142/9789814324359_0162 fatcat:ss7ezul2jjemhnxhnl3jjzhwzq

Algebraic Soft-Decision Decoding of Reed-Solomon Codes with Erasures on Gaussian Channels

F.R. Coutinho, E.M.G. Fernández
2007 Journal of Communication and Information Systems  
The search for increasing the error performance of algebraic soft-decision decoding of high rate Reed-Solomon (RS) codes motivates the development of this work in an attempt to determine the ultimate error-correcting  ...  capabilities of algebraic soft-decision decoding of RS codes in Gaussian channels with erasures.  ...  by changing the conventional bounded distance decoding paradigm to a list decoding procedure.  ... 
doi:10.14209/jcis.2007.3 fatcat:bbpfxruyofhf5blg7dg7r2uvri

Algebraic geometry codes and some applications [article]

Alain Couvreur, Hugues Randriambololona
2020 arXiv   pre-print
We summarize the major results on various problems such as: asymptotic parameters, improved estimates on the minimum distance, and decoding algorithms.  ...  This article surveys the development of the theory of algebraic geometry codes since their discovery in the late 70's.  ...  (1) Beware that the terminology on Reed-Solomon codes varies in the literature with several names for variants: generalized Reed-Solomon codes, extended Reed-Solomon codes, doubly extended Reed-Solomon  ... 
arXiv:2009.01281v1 fatcat:2o7m2zhylnanjpljkikw4sx2ya

Decoding of Interleaved Reed Solomon Codes over Noisy Data [chapter]

Daniel Bleichenbacher, Aggelos Kiayias, Moti Yung
2003 Lecture Notes in Computer Science  
We propose a new decoding algorithm for interleaved Reed-Solomon Codes that attempts to correct all "interleaved" codewords simultaneously.  ...  Our construction yields a decoding algorithm for interleaved RS-codes that allows efficient transmission arbitrarily close to the channel capacity in the NBSC model.  ...  The authors wish to thank Alexander Barg for helpful discussions.  ... 
doi:10.1007/3-540-45061-0_9 fatcat:5eoffdsxefcq5j4zpffckkshqu

Some Applications of Coding Theory in Computational Complexity [article]

Luca Trevisan
2004 arXiv   pre-print
In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography.  ...  Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs.  ...  In Fall 2003 I taught a graduate class on applications of coding theory to computational complexity, I thank the students who took the class for their enthusiasm and participation.  ... 
arXiv:cs/0409044v1 fatcat:46gkw3tgnfbz7gfkwyrzgbw4wm

Folded Algebraic Geometric Codes From Galois Extensions [article]

Ming-Deh Huang, Anand Kumar Narayanan
2009 arXiv   pre-print
We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm.  ...  This work is an extension of Folded Reed Solomon codes to the setting of Algebraic Geometric codes.  ...  Reed Solomon codes with unique decoding can correct a fraction of 1− R 2 errors. The Guruswami-Sudan List Decoding algorithm for Reed Solomon codes improved the bound to δ = 1 − √ R [11] .  ... 
arXiv:0901.1162v1 fatcat:4457m6w5ljfdlaahkacikihwsa

List Decoding Group Homomorphisms Between Supersolvable Groups

Alan Guo, Madhu Sudan, Marc Herbstritt
2014 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We show that the set of homomorphisms between two supersolvable groups can be locally list decoded up to the minimum distance of the code, extending the results of Dinur et al who studied the case where  ...  Moreover, when specialized to the abelian case, our proof is more streamlined and gives a better constant in the exponent of the list size. The constant is improved from about 3.5 million to 105.  ...  In particular, one abstraction of the (original) Goldreich-Levin algorithm is that it uses coding theory (in particular, the Johnson bound of coding theory) to get a combinatorial bound on the list size  ... 
doi:10.4230/lipics.approx-random.2014.737 dblp:conf/approx/GuoS14 fatcat:dlsnd24zjbdxti2tzfxgswceou

List decoding group homomorphisms between supersolvable groups [article]

Alan Guo, Madhu Sudan
2014 arXiv   pre-print
We show that the set of homomorphisms between two supersolvable groups can be locally list decoded up to the minimum distance of the code, extending the results of Dinur et al who studied the case where  ...  Moreover, when specialized to the abelian case, our proof is more streamlined and gives a better constant in the exponent of the list size. The constant is improved from about 3.5 million to 105.  ...  Thus the abstraction and generalization improves the list-size bounds even in the abelian case. Our analysis shows that the list-decoding radius is as large as the distance.  ... 
arXiv:1404.4273v1 fatcat:t2kvapripbefxc66m2b4bcykcm

Efficiently decoding Reed-Muller codes from random errors

Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk
2016 Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016  
algorithm for decoding Reed-Solomon codes.  ...  The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding Reed-Muller codes that are based on the recursive  ...  We thank Venkatesan Guruswami and anonymous reviewers for pointing out the abstraction of Algorithm 1 given in Section 3.  ... 
doi:10.1145/2897518.2897526 dblp:conf/stoc/SaptharishiSV16 fatcat:7usy2ucyvndmbi6xiuwv52qyly

Efficiently Decoding Reed–Muller Codes From Random Errors

Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk
2017 IEEE Transactions on Information Theory  
algorithm for decoding Reed-Solomon codes.  ...  The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding Reed-Muller codes that are based on the recursive  ...  We thank Venkatesan Guruswami and anonymous reviewers for pointing out the abstraction of Algorithm 1 given in Section 3.  ... 
doi:10.1109/tit.2017.2671410 fatcat:k2y4eqqrsfhqhejjc6tzthmkdm
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