On the Performance of Multivariate Interpolation Decoding of Reed-Solomon Codes

Parvaresh, Farzad; Taghavi, Mohammad Hossein; Vardy, Alexander

doi:10.1109/isit.2006.261905

Cited by 6 publications

(17 citation statements)

References 8 publications

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“…In the case of two variables and high rates the complexity we obtain in O n 2 m 4 is better than the complexity in O n 2 m 5 obtained in ( [7], [12]). The general complexity we obtain for our algorithm is also better than the complexity O n 2 m 3M −1 (n/k) M −1 M of the generalization [13] for three and more variables of the algorithm of [7].…”

Section: Introductionmentioning

confidence: 84%

“…These important results induced much work to optimize algorithms and fast algorithms for bivariate Hermite interpolation ( [14], [11], [3], [10], [1]) and more recently for multivariate Hermite interpolation ( [12], [13]). …”

Section: Introductionmentioning

confidence: 99%

“…Notice that for very small dimensions k, the complexity increases, this case may be of interest for some cryptographic applications where it may be relevant to decode Reed-Solomon codes with small rates [6]. This interpolation algorithm was recently generalized to three variables and more by Parvaresh and Vardy in ( [12], [13]). …”

Section: Introductionmentioning

confidence: 99%

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Improved Hermite multivariate polynomial interpolation

Gaborit

Ruatta

2006

2006 IEEE International Symposium on Information Theory

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In this paper we give an algorithm with complexity O`µ 2´t o solve Hermite multivariate polynomial interpolation with µ conditions on its Hasse derivatives. In the case of bivariate interpolation used to perform list-decoding on Reed-Solomon of length n and dimension k with multiplicity m on each point, it permits to obtain a complexity in O`n 2 m 4´w hich does not depend on the rate k/n and better than previously known complexity in O " n 2 m 5 (n/k). This algorithm can also be used for recent interpolation list-decoding with three and more variables. For interpolation on polynomial with n points and M variables with prescribed multiplication order m the general complexity of the algorithm is O`n 2 m 2M´.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

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Improved Hermite multivariate polynomial interpolation

Gaborit

Ruatta

2006

2006 IEEE International Symposium on Information Theory

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“…Herein, we move on to interpolation decoding in M+1 variables, where M 1 is the integer parameter in Theorem 1. Much of this framework was developed in our earlier papers [18,19]. However, since that work is not yet readily accessible, we will give a brief primer on multivariate interpolation decoding in the next section.…”

Section: The Underlying Ideasmentioning

confidence: 99%

“…Rather than trying to devise a better list-decoder for Reed-Solomon codes (which, as it turns out [19], is what all the previous papers attempted to do), we construct better codes. Our construction is still based upon the Reed-Solomon codes, yet differs from them in an essential way.…”

Section: The Underlying Ideasmentioning

confidence: 99%

Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time

Parvaresh

Vardy

46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)

138

167

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We introduce a new family of error-correcting codes that have a polynomial-time encoder and a polynomial-time listdecoder, correcting a fraction of adversarial errors up towhere R is the rate of the code and M 1 is an arbitrary integer parameter. This makes it possible to decode beyond the Guruswami-Sudan radius of 1 − √ R for all rates less than 1/16. Stated another way, for any ε > 0, we can listdecode in polynomial time a fraction of errors up to 1 − ε with a code of length n and rate Ω ε/log(1/ε) , defined over an alphabet of size n M = n O(log(1/ε)) . Notably, this error-correction is achieved in the worst-case against adversarial errors: a probabilistic model for the error distribution is neither needed nor assumed. The best results so far for polynomial-time list-decoding of adversarial errors required a rate of O(ε 2 ) to achieve the correction radius of 1−ε.Our codes and list-decoders are based on two key ideas. The first is the transition from bivariate polynomial interpolation, pioneered by Sudan and Guruswami-Sudan [12,22], to multivariate interpolation decoding. The second idea is to part ways with Reed-Solomon codes, for which numerous prior attempts [2,3,12,18] at breaking the O(ε 2 ) rate barrier in the worst-case were unsuccessful. Rather than devising a better list-decoder for Reed-Solomon codes, we devise better codes. Standard Reed-Solomon encoders view a message as a polynomial f (X) over a field F q , and produce the corresponding codeword by evaluating f (X) at n distinct elements of F q . Herein, given f (X), we first compute one or more related polynomials g 1 (X), g 2 (X), . . . , g M−1 (X) and produce the corresponding codeword by evaluating all these polynomials. Correlation between f (X) and g i (X), carefully designed into our encoder, then provides the additional information we need to recover the encoded message from the output of the multivariate interpolation process.

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Collaborative algebraic decoding of interleaved Reed–Solomon codes

Shayegh

Soleymani

2011

Trans. Emerging Tel. Tech.

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We derive and analyse an algorithm for collaborative decoding of heterogeneous interleaved Reed–Solomon (IRS) codes. They are generated by interleaving several codewords from different Reed–Solomon codes with the same length over the same Galois field. The basis of the decoding algorithm is similar to the Guruswami–Sudan (GS) decoding method. However, here multivariate interpolation is used to decode all the codewords of the interleaved scheme simultaneously. In the presence of burst errors, we show that the error‐correction capability of this algorithm is larger than that of independent decoding of each codeword using the standard GS method. In the latter case, the error‐correction capability is equal to the decoding radius of the GS algorithm for the Reed–Solomon code with the largest dimension. Also, concatenated codes using IRS codes as their outer codes and binary linear block codes as their inner codes are considered. Assuming maximum likelihood decoding of the inner code, we derive upper and lower bounds for the word error probability of concatenated codes over additive white Gaussian noise channel with binary phase‐shift keying modulation for both cases of independent and collaborative decoding of the outer IRS codes. We show that collaborative decoding provides considerable coding gain compared with independent decoding. Copyright © 2011 John Wiley & Sons, Ltd.

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On the Performance of Multivariate Interpolation Decoding of Reed-Solomon Codes

Cited by 6 publications

References 8 publications

Improved Hermite multivariate polynomial interpolation

Improved Hermite multivariate polynomial interpolation

Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time

Collaborative algebraic decoding of interleaved Reed–Solomon codes

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