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

Guruswami, Venkatesan; Patthak, Anindya C.

doi:10.1109/focs.2006.23

Cited by 10 publications

(28 citation statements)

References 14 publications

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“…(Essentially, the freedom to do -variate interpolation for a parameter of our choosing allows us to work with simple interpolation, while still gaining in error-correction radius with increasing . This phenomenon also occurred in one of the algorithms in [12] for list-decoding correlated algebraic-geometric codes. )…”

Section: B Welch-berlekamp Style Interpolationmentioning

confidence: 88%

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Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes

Guruswami

Wang

2013

IEEE Trans. Inform. Theory

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Folded Reed-Solomon (RS) codes are an explicit family of codes that achieve the optimal tradeoff between rate and list error-correction capability: specifically, for any , Guruswami and Rudra presented an time algorithm to list decode appropriate folded RS codes of rate from a fraction of errors. The algorithm is based on multivariate polynomial interpolation and root-finding over extension fields. It was noted by Vadhan that interpolating a linear polynomial suffices for a statement of the above form. Here, we give a simple linear-algebra-based analysis of this variant that eliminates the need for the computationally expensive root-finding step over extension fields (and indeed any mention of extension fields). The entire list-decoding algorithm is linear-algebraic, solving one linear system for the interpolation step, and another linear system to find a small subspace of candidate solutions. Except for the step of pruning this subspace, the algorithm can be implemented to run in quadratic time. We also consider a closely related family of codes, called (order ) derivative codes and defined over fields of large characteristic, which consist of the evaluations of as well as its first formal derivatives at distinct field elements. We show how our linear-algebraic methods for folded RS codes can be used to show that derivative codes can also achieve the above optimal tradeoff. The theoretical drawback of our analysis for folded RS codes and derivative codes is that both the decoding complexity and proven worst-case list-size bound are . By combining the above idea with a pseudorandom subset of all polynomials as messages, we get a Monte Carlo construction achieving a list-size bound of which is quite close to the existential bound (however, the decoding complexity remains ). Our work highlights that constructing an explicit subspace-evasive subset that has small intersection with low-dimensional subspaces-an interesting problem in pseudorandomness in its own right-could lead to explicit codes with better list-decoding guarantees.

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Section: B Welch-berlekamp Style Interpolationmentioning

confidence: 88%

“…(We know that the degree of for is at most , so when and , but for notational ease let us introduce these coefficients.) For , define the polynomial (12) We know that , and therefore . By Condition (11), for each , the coefficient of in the polynomial equals 0.…”

Section: Retrieving Candidate Polynomialsmentioning

confidence: 99%

Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes

Guruswami

Wang

2013

IEEE Trans. Inform. Theory

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“…The complexity of the algorithm is polynomial assuming availability of a polynomial amount of pre-processed information about the code [43]. For a family of AG codes that achieve the best rate vs. distance trade-off, it was recently shown how to compute the required pre-processed information in polynomial time [36].…”

Section: Related Results On Algebraic List Decodingmentioning

confidence: 99%

“…Another potential avenue for improving the complexity and list size is via a generalization of folded RS codes to folded algebraicgeometric (AG) codes. In [36], the authors define correlated AG codes, and describe list decoding algorithms for those codes, based on a generalization of the Parvaresh-Vardy approach to the general class of algebraic-geometric codes (of which RS codes are a special case). However, to relate folded AG codes to correlated AG codes like we did for RS codes requires bijections on the set of rational points of the underlying algebraic curve that have some special, hard to guarantee, property.…”

Section: Improving List Size Of Capacity-achieving Codesmentioning

confidence: 99%

Algorithmic Results in List Decoding

Guruswami

2006

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Error-correcting codes are used to cope with the corruption of data by noise during communication or storage. A code uses an encoding procedure that judiciously introduces redundancy into the data to produce an associated codeword. The redundancy built into the codewords enables one to decode the original data even from a somewhat distorted version of the codeword. The central trade-off in coding theory is the one between the data rate (amount of non-redundant information per bit of codeword) and the error rate (the fraction of symbols that could be corrupted while still enabling data recovery). The traditional decoding algorithms did as badly at correcting any error pattern as they would do for the worst possible error pattern. This severely limited the maximum fraction of errors those algorithms could tolerate. In turn, this was the source of a big hiatus between the error-correction performance known for probabilistic noise models (pioneered by Shannon) and what was thought to be the limit for the more powerful, worst-case noise models (suggested by Hamming).In the last decade or so, there has been much algorithmic progress in coding theory that has bridged this gap (and in fact nearly eliminated it for codes over large alphabets). These developments rely on an error-recovery model called "list decoding," wherein for the pathological error patterns, the decoder is permitted to output a small list of candidates that will include the original message. This book introduces and motivates the problem of list decoding, and discusses the central algorithmic results of the subject, culminating with the recent results on achieving "list decoding capacity." The potential of list decoding 113

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“…Contained in [10] is a scheme to reduce the alphabet size based on concatenating Folded Reed Solomon codes with appropriate inner codes. Guruswami and Pathak [9] provide a generalization of the Parvaresh-Vardy code to the Algebraic-Geometric setting thereby reducing the alphabet size. By generalizing Folded Reed Solomon codes to Folded Algebraic Geometric codes we present a purely algebraic means of achieving the rate error correction tradeoff with alphabet size independent of the block length.…”

Section: Introductionmentioning

confidence: 99%

Folded algebraic-geometric codes from Galois extensions

Huang¹,

Narayanan²

2010

Finite Fields: Theory and Applications

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We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. 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. This work is an extension of Folded Reed Solomon codes to the setting of Algebraic Geometric codes. We describe two constructions based on this framework depending on if the order of the automorphism used to fold the code is large or small compared to the block length. When the automorphism is of large order, the codes have polynomially bounded list size in the worst case. This construction gives codes of rate R over an alphabet of size independent of block length that can correct a fraction of 1−R −ǫ errors subject to the existence of asymptotically good towers of function fields with large automorphisms. The second construction addresses the case when the order of the element used to fold is small compared to the block length. In this case a heuristic analysis shows that for a random received word, the expected list size and the running time of the decoding algorithm are bounded by a polynomial in the block length. When applied to the GarciaStichtenoth tower, this yields codes of rate R over an alphabet of size ( , that can correct a fraction of 1 − R − ǫ errors.

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

Cited by 10 publications

References 14 publications

Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes

Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes

Algorithmic Results in List Decoding

Folded algebraic-geometric codes from Galois extensions

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