Decoding Reed-Solomon Codes Beyond Half the Minimum Distance

Nielsen, R. Refslund; Høholdt, Tom

doi:10.1007/978-3-642-57189-3_20

Cited by 61 publications

(71 citation statements)

References 4 publications

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“…Heuristically, this fact could be related to [40,Proposition 12,page 9] which states that the probability of having more than one codeword in a Hamming ball, whose radius corresponds to the Sudan algorithm decoding radius, is close to zero. The degree d of F is related to the number c of codewords within the Hamming ball by c d. And, in practice, we observe that d is close to 1 when c = 1 with probability close to 1.…”

Section: Example 6 With R = ◗[[T]] Here Is the Trace Of Algorithm 5 mentioning

confidence: 99%

Polynomial root finding over local rings and application to error correcting codes

Berthomieu

Lecerf

Quintin

2013

AAECC

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Section: Example 6 With R = ◗[[T]] Here Is the Trace Of Algorithm 5 mentioning

confidence: 99%

Polynomial root finding over local rings and application to error correcting codes

Berthomieu

Lecerf

Quintin

2013

AAECC

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“…For this instance, Wu's algorithm using (22) computes s = 2, which is also the minimum multiplicity. Now Wu's algorithm computes M = 5 and ρ = 72 using (23) and (24), respectively. With these values, Wu's algorithm requires solving a system of N = 381 homogeneous equations involving U = 408 unknowns.…”

Section: Complexitymentioning

confidence: 99%

“…, n, with multiplicity s is an ideal I s . From this observation several authors including Alekhnovich [2], Nielsen and Høholdt [23], Kuijper and Polderman [16], O'Keeffe and Fitzpatrick [24], and Lee and O'Sullivan [19], formulated the interpolation step of the list decoding algorithm as the problem of finding the minimal weight polynomial from the ideal I s . Clearly the minimal weight polynomial will appear as the minimal polynomial in a minimal Gröbner basis of I s computed with respect to the corresponding weighted term order.…”

Section: Introductionmentioning

confidence: 99%

A Parametric Approach to List Decoding of Reed-Solomon Codes Using Interpolation

Ali

Kuijper

2011

IEEE Trans. Inform. Theory

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In this paper we present a minimal list decoding algorithm for Reed-Solomon (RS) codes. Minimal list decoding for a code C refers to list decoding with radius L, where L is the minimum of the distances between the received word r and any codeword in C. We consider the problem of determining the value of L as well as determining all the codewords at distance L. Our approach involves a parametrization of interpolating polynomials of a minimal Gröbner basis G. We present two efficient ways to compute G. We also show that so-called re-encoding can be used to further reduce the complexity. We then demonstrate how our parametric approach can be solved by a computationally feasible rational curve fitting solution from a recent paper by Wu. Besides, we present an algorithm to compute the minimum multiplicity as well as the optimal values of the parameters associated with this multiplicity which results in overall savings in both memory and computation.

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“…Each such polynomial is placed on the list. A solution to the interpolation problem exists if is strictly less than the number of monomials in such that is of minimal weighted degree [24]. A sufficient condition for a codeword to be on the GS generated list is [7], [9] (2) 1 To prevent notational ambiguity, kxk will denote the magnitude of x.…”

Section: Algebraic Soft Decodingmentioning

confidence: 99%

Iterative algebraic soft-decision list decoding of Reed-Solomon codes

El-Khamy

McEliece

2006

IEEE J. Select. Areas Commun.

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Abstract-In this paper, we present an iterative soft-decision decoding algorithm for Reed-Solomon (RS) codes offering both complexity and performance advantages over previously known decoding algorithms. Our algorithm is a list decoding algorithm which combines two powerful soft-decision decoding techniques which were previously regarded in the literature as competitive, namely, the Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation based on adaptive parity-check matrices, recently proposed by Jiang and Narayanan. Building on the Jiang-Narayanan algorithm, we present a belief-propagation-based algorithm with a significant reduction in computational complexity. We introduce the concept of using a belief-propagation-based decoder to enhance the soft-input information prior to decoding with an algebraic soft-decision decoder. Our algorithm can also be viewed as an interpolation multiplicity assignment scheme for algebraic soft-decision decoding of RS codes.

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Decoding Reed-Solomon Codes Beyond Half the Minimum Distance

Cited by 61 publications

References 4 publications

Polynomial root finding over local rings and application to error correcting codes

Polynomial root finding over local rings and application to error correcting codes

A Parametric Approach to List Decoding of Reed-Solomon Codes Using Interpolation

Iterative algebraic soft-decision list decoding of Reed-Solomon codes

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