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Expanderbased constructions of efficiently decodable codes
 In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science
, 2001
"... We present several novel constructions of codes which share the common thread of using expander (or expanderlike) graphs as a component. The expanders enable the design of efficient decoding algorithms that correct a large number of errors through various forms of “voting ” procedures. We consider ..."
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We present several novel constructions of codes which share the common thread of using expander (or expanderlike) graphs as a component. The expanders enable the design of efficient decoding algorithms that correct a large number of errors through various forms of “voting ” procedures. We consider both the notions of unique and list decoding, and in all cases obtain asymptotically good codes which are decodable up to a “maximum” possible radius and either (a) achieve a similar rate as the previously best known codes but come with significantly faster algorithms, or (b) achieve a rate better than any prior construction with similar errorcorrection properties. Among our main results are: ¯ Codes of rate ª � over constantsized alphabet that can be list decoded in quadratic time from � errors. This matches the performance of the best algebraicgeometric (AG) codes, but with much faster encoding and decoding algorithms. ¯ Codes of rate ª � over constantsized alphabet that can be uniquely decoded from � � errors in nearlinear time (once again this matches AGcodes with much faster algorithms). This construction is similar to that of [1], and our decoding algorithm can be viewed as a positive resolution of their main open question. ¯ Lineartime encodable and decodable binary codes of positive rate 1 (in fact, rate ª � � ) that can correct up to � � � fraction errors. Note that this is the best errorcorrection one can hope for using unique decoding of binary codes. This significantly improves the fraction of errors corrected by the earlier lineartime codes of Spielman [19] and the lineartime decodable codes of [18, 22].
Ideal errorcorrecting codes: Unifying algebraic and numbertheoretic algorithms
 NOTES IN COMP. SCI
, 2001
"... Over the past five years a number of algorithms decoding some wellstudied errorcorrecting codes far beyond their “errorcorrecting radii” have been developed. These algorithms, usually termed as listdecoding algorithms, originated with a listdecoder for ReedSolomon codes [36, 17], and were soon ..."
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Over the past five years a number of algorithms decoding some wellstudied errorcorrecting codes far beyond their “errorcorrecting radii” have been developed. These algorithms, usually termed as listdecoding algorithms, originated with a listdecoder for ReedSolomon codes [36, 17], and were soon extended to decoders for Algebraic Geometry codes [33, 17] and as also some numbertheoretic codes [12, 6, 16]. In addition to their enhanced decoding capability, these algorithms enjoy the benefit of being conceptually simple, fairly general [16], and are capable of exploiting softdecision information in algebraic decoding [24]. This article surveys these algorithms and highlights some of these features.
Correlated AlgebraicGeometric codes: Improved list decoding over bounded alphabets
 Mathematics of Computation
"... Abstract. We define a new family of errorcorrecting codes based on algebraic curves over finite fields, and develop efficient list decoding algorithms for them. Our codes extend the class of algebraicgeometric (AG) codes via a (nonobvious) generalization of the approach in the recent breakthrough ..."
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Cited by 5 (5 self)
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Abstract. We define a new family of errorcorrecting codes based on algebraic curves over finite fields, and develop efficient list decoding algorithms for them. Our codes extend the class of algebraicgeometric (AG) codes via a (nonobvious) generalization of the approach in the recent breakthrough work of Parvaresh and Vardy (2005). Our work shows that the PV framework applies to fairly general settings by elucidating the key algebraic concepts underlying it. Also, more importantly, AG codes of arbitrary block length exist over fixed alphabets Σ, thus enabling us to establish new tradeoffs between the list decoding radius and rate over a bounded alphabet size. 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 tradeoff between rate and fraction of errors corrected) over large alphabets. A similar extension of this work along the lines of Guruswami and Rudra could have substantial impact. Indeed, it could give better tradeoffs than currently known over a fixed alphabet (say, GF(212)), which in turn, upon concatenation with a fixed, wellunderstood binary code, could take us closer to the list decoding capacity for binary codes. This may also be a promising way to address the significant complexity drawback of the result of Guruswami and Rudra, and to enable approaching capacity with bounded list size independent of the block length (the list size and decoding complexity in their work are both nΩ(1/ε) where ε is the distance to capacity). Similar to algorithms for AG codes from Guruswami and Sudan (1999) and (2001), our encoding/decoding algorithms run in polynomial time assuming a natural polynomialsize representation of the code. For codes based on a specific “optimal ” algebraic curve, we also present an expected polynomial time algorithm to construct the requisite representation. This in turn fills an important void in the literature by presenting an efficient construction of the representation often assumed in the list decoding algorithms for AG codes. 1.
List Decoding of AlgebraicGeometric Codes
, 2001
"... We generalize the list decoding algorithm for onepoint (strongly) algebraicgeometric codes by Guruswami and Sudan to all algebraicgeometric codes. Moreover, our algorithm works for a generalized Hamming distance with real weight coefficients rather than integer weight coefficients. This is more s ..."
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We generalize the list decoding algorithm for onepoint (strongly) algebraicgeometric codes by Guruswami and Sudan to all algebraicgeometric codes. Moreover, our algorithm works for a generalized Hamming distance with real weight coefficients rather than integer weight coefficients. This is more suitable for softdecision decoding where these weight coefficients are, for instance, absolute values of loglikelihoods. Given a vector y 2 F n
Decoding Concatenated Codes using Soft Information
 In Proceedings of the 17th IEEE Conference of Computational Complexity
, 2001
"... We present a decoding algorithm for concatenated codes when the outer code is a ReedSolomon code and the inner code is arbitrary. "Soft" information on the reliability of various symbols is passed by the inner decodings and exploited in the ReedSolomon decoding. This is the first analy ..."
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Cited by 4 (2 self)
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We present a decoding algorithm for concatenated codes when the outer code is a ReedSolomon code and the inner code is arbitrary. "Soft" information on the reliability of various symbols is passed by the inner decodings and exploited in the ReedSolomon decoding. This is the first analysis of such a soft algorithm that works for arbitrary inner codes; prior analyses could only handle some special inner codes. Crucial to our analysis is a combinatorial result on the coset weight distribution of codes given only its minimum distance. Our result enables us to decode essentially up to the "Johnson radius" of a concatenated code when the outer distance is large (the Johnson radius is the "a priori list decoding radius" of a code as a function of its distance). As a consequence, we are able to present simple and efficient constructions of qary linear codes that are list decodable up to a fraction (1 1=q ") of errors and have rate " 6 ). The previous constructions of linear codes with a similar rate used algebraicgeometric codes and thus suffered from a complicated construction and slow decoding.