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

Dept. of Electr. & Electron. Eng., Univ. of Melbourne, Melbourne, VIC, Australia
IEEE Transactions on Information Theory (Impact Factor: 2.33). 11/2011; DOI: 10.1109/TIT.2011.2165803
Source: IEEE Xplore


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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    • "In this section, we give some general definitions and results on L q (x, q m ) ℓ , including the Predictable Leading Monomial property. All of these are analogous to the definitions and results for modules over F q m [x] (equipped with normal polynomial multiplication) from [2], see also the early work by Fitzpatrick [4]. To our knowledge , this generalization to linearized polynomials with multiplication replaced by composition, has not been presented before in the literature—we find that it leads to powerful and flexible tools that lead to straightforward proofs for rank-metric decoding. "
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    ABSTRACT: We show how Gabidulin codes can be list decoded by using a parametrization approach. Our decoding algorithm computes a list of all closest codewords to a given received word. We consider a certain module, called the interpolation module, over the ring of linearized polynomials with respect to composition of polynomials. The Predictable Leading Monomial property for minimal bases of this interpolation module is stated and proved, which is then used as a key ingredient for our parametrization. The parametrization is based on a minimal basis for the interpolation module, which is why we furthermore formulate two subalgorithms, one using the extended Euclidean algorithm and an iterative one, for finding such a basis.
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    • "First, as there may exist interpolation points at infinity, we can not use the traditional module based bivariate interpolation algorithm (including Kötter algorithm[11] and Lee-O'Sullivan algorithm[12]) to do the interpolation because the legal interpolation result set with particular y degree limit is not a module. Second, if we can not guarantee u(0) = 0, we can not get Taylor series of v/u at zero by Roth-Ruckenstein algorithm[8] and Padé approximation can not be used either, which is not mentioned in some previous result [9], [10]. The following result is organized as follows. "
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    ABSTRACT: In this paper, a new algebraic soft-decision decoding algorithm for Reed-Solomon code is presented. It is based on rational interpolation and the interpolation points are constructed by Berlykamp-Messay algorithm. Unlike the traditional K{\"o}tter-Vardy algorithm, new algorithm needs interpolation for two smaller multiplicity matrixes, due to the corresponding factorization algorithm for re-constructing codewords.
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    • "In our algorithm we construct, via a simple update matrix, a minimal Gröbner basis at each step. This setup allows for straightforward conclusions on minimality and parametrization due to the Predictable Leading Monomial Property, as in [1] and [6]. "
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    ABSTRACT: We show how Gabidulin codes can be list decoded by using an iterative parametrization approach. For a given received word, our decoding algorithm processes its entries one by one, constructing four polynomials at each step. This then yields a parametrization of interpolating solutions for the data so far. From the final result a list of all codewords that are closest to the received word with respect to the rank metric is obtained.
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