Efficient Frequency-Domain Decoding Algorithms for Reed-Solomon Codes (original) (raw)
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FFT Algorithm for Binary Extension Finite Fields and Its Application to Reed–Solomon Codes
IEEE Transactions on Information Theory, 2016
Recently, a new polynomial basis over binary extension fields was proposed such that the fast Fourier transform (FFT) over such fields can be computed in the complexity of order O(n lg(n)), where n is the number of points evaluated in FFT. In this work, we reformulate this FFT algorithm such that it can be easier understood and be extended to develop frequencydomain decoding algorithms for (n = 2 m , k) systematic Reed-Solomon (RS) codes over F2m , m ∈ Z + , with n − k a power of two. First, the basis of syndrome polynomials is reformulated in the decoding procedure so that the new transforms can be applied to the decoding procedure. A fast extended Euclidean algorithm is developed to determine the error locator polynomial. The computational complexity of the proposed decoding algorithm is O(n lg(n − k) + (n − k) lg 2 (n − k)), improving upon the best currently available decoding complexity O(n lg 2 (n) lg lg(n)), and reaching the best known complexity bound that was established by Justesen in 1976. However, Justesen's approach is only for the codes over some specific fields, which can apply Cooley-Tucky FFTs. As revealed by the computer simulations, the proposed decoding algorithm is 50 times faster than the conventional one for the (2 16 , 2 15) RS code over F 2 16 .
Novel Polynomial Basis With Fast Fourier Transform and Its Application to Reed–Solomon Erasure Codes
IEEE Transactions on Information Theory, 2016
In this paper, we present a fast Fourier transform (FFT) algorithm over extension binary fields, where the polynomial is represented in a non-standard basis. The proposed Fourier-like transform requires O(h lg(h)) field operations, where h is the number of evaluation points. Based on the proposed Fourier-like algorithm, we then develop the encoding/decoding algorithms for (n = 2 m , k) Reed-Solomon erasure codes. The proposed encoding/erasure decoding algorithm requires O(n lg(n)), in both additive and multiplicative complexities. As the complexity leading factor is small, the proposed algorithms are advantageous in practical applications. Finally, the approaches to convert the basis between the monomial basis and the new basis are proposed.
Two new decoding algorithms for Reed-Solomon codes
Applicable Algebra in Engineering, Communication and Computing, 1994
The subject of decoding Reed-Solomon codes is considered, By reformulating the Berlekamp and Welch key equation and introducing new versions of this key equation, two new decoding algorithms for Reed-Solomon codes will be presented. The two new decoding algorithms are significant for three reasons. Firstly the new equations and algorithms represent a novel approach to the extensively researched problem of decoding Reed-Solomon codes. Secondly the algorithms have algorithmic and implementation complexity comparable to existing decoding algorithms, and as such present a viable solution for decoding Reed-Solomon codes. Thirdly the new ideas presented suggest a direction for future research. The first algorithm uses the extended Euclidean algorithm an~t is very efficient for a systolic VLSI implementation. The second decoding algorithm presented is similar in nature to the original decoding algorithm of Peterson except that the syndromes do not need to be computed and the remainders are used directly. It has a regular structure and will be efficient for implementation only for correcting a small number of errors. A systolic design for computing the Lagrange interpolation of a polynomial, which is needed for the first decoding algorithm, is also presented.
A Transform-Domain Decoding Algorithm for Reed-Solomon Codes
2006 IEEE International Conference on Ultra-Wideband, 2006
This paper presents a Reed-Solomon decoding algorithm based on the Euclidean algorithm. The algorithm is conceptually simple and operates only in transform domain. The spectrum of the codeword is directly computed without the explicit knowledge of error-locator and error-evaluator polynomials; the Chien search and the Forney algorithm are not necessary.
Decoding of Reed Solomon Codes beyond the Error-Correction Bound
Journal of Complexity, 1997
We present a randomized algorithm which takes as input n distinct points f(x ; y )g from F 2 F (where F is a field) and integer parameters t and d and returns a list of all univariate polynomials f over F in the variable x of degree at most d which agree with the given set of points in at least t places (i.e., y = f (x ) for at least t values of i), provided t = ( p nd). The running time is bounded by a polynomial in n. This immediately provides a maximum likelihood decoding algorithm for Reed Solomon Codes, which works in a setting with a larger number of errors than any previously known algorithm. To the best of our knowledge, this is the first efficient (i.e., polynomial time bounded) algorithm which provides error recovery capability beyond the error-correction bound of a code for any efficient (i.e., constant or even polynomial rate) code.
IEEE Transactions on Communications, 2001
The central problem in the implementation of a Reed-Solomon code is finding the roots of the error locator polynomial. In 1967, Berlekamp et al. found an algorithm for finding the roots of an affine polynomial in GF(2 m) that can be used to solve this problem. In this paper, it is shown that this Berlekamp-Rumsey-Solomon algorithm, together with the Chien-search method, makes possible a fast decoding algorithm in the standard-basis representation that is naturally suitable in a software implementation. Finally, simulation results for this fast algorithm are given.
Fast parallel algorithms for decoding Reed-Solomon codes based on remainder polynomials
Information Theory, IEEE Transactions on, 1995
The problem of decoding cyclic error correcting codes is one of solving a constrained polynomial congruence, often achieved using the BerlekampMassey or the extended Euclidean algorithm on a key equation involving the syndrome polynomial. A module-theoretic approach to the solution of polynomial congruences is developed here using the notion of exact sequences. This technique is applied to the Welch-Berlekamp key equation for decoding ReedSolomon codes for which the computation of syndromes is not required. It leads directly to new and efficient parallel decoding algorithms that can be realized with a systolic array. The architectural issues for one of these parallel decoding algorithms are examined in some detail. Index Tenns-ReedSolomon codes, decoding algorithms, systolic arrays, Welch-Berlekamp equations, modules.
Fast transform decoding of nonsystematic reed-solomon codes
IEE Proceedings E Computers and Digital Techniques, 1990
This article considers a Reed-Solomon (RS) code to be a special case of a redundant residue polynomial (RRP) code, and presents a fast transform decoding algorithm to correct both errors and erasures, This decoding scheme is an improvement of the decoding algorithm for the RRP code suggested by Shiozaki and Nishida [1], and can be realized readily on VLSI chips.
A Fast Algorithm for the Syndrome Calculation in Algebraic Decoding of Reed–Solomon Codes
IEEE Transactions on Communications, 2007
In this letter, Fedorenko and Trifonov's procedure is applied to evaluate the syndrome of the received word in timedomain Reed-Solomon decoders. This application leads to a substantial reduction of the computational complexity of the syndrome polynomial for correcting both errors and erasures. Moreover, simulation results for this new syndrome method are given.
Fast Ecoding/Decoding Algorithms for Reed-Solomon Erasure Codes
In this paper, we present a new basis of polynomial over finite fields of characteristic two and then apply it to the encoding/decoding of Reed-Solomon erasure codes. The proposed polynomial basis allows that h-point polynomial evaluation can be computed in O(h log 2 (h)) finite field operations with small leading constant. As compared with the canonical polynomial basis, the proposed basis improves the arithmetic complexity of addition, multiplication, and the determination of polynomial degree from O(h log 2 (h) log 2 log 2 (h)) to O(h log 2 (h)). Based on this basis, we then develop the encoding and erasure decoding algorithms for the (n = 2 r , k) Reed-Solomon codes. Thanks to the efficiency of transform based on the polynomial basis, the encoding can be completed in O(n log 2 (k)) finite field operations, and the erasure decoding in O(n log 2 (n)) finite field operations. To the best of our knowledge, this is the first approach supporting Reed-Solomon erasure codes over characteristic-2 finite fields while achieving a complexity of O(n log 2 (n)), in both additive and multiplicative complexities. As the complexity leading factor is small, the algorithms are advantageous in practical applications.