Two new decoding algorithms for Reed-Solomon codes (original) (raw)
Related papers
IEEE Transactions on Communications, 2011
It is well-known that the Euclidean algorithm can be used o find the systematic errata-locator polynomial and the errata-evaluator polynomial simultaneously in Berlekamp's key equation that is needed to decode a Reed-Solomon (RS) codes. In this paper, a simplified decoding algorithm to correct both errors and erasures is used in conjunction with the Euclidean algorithm for efficiently decoding nonsystematic RS codes. In fact, this decoding algorithm is an appropriate modification to the algorithm developed by Shiozaki and Gao. Based on the ideas presented above, a fast algorithm described from Blahut's classic book is derivated and proved in this paper to correct erasures as well as errors by replacing the Euclidean algorithm by the Berlekamp-Massey (BM) algorithm. These facts lead to significantly reduce the decoding complexity of the proposed RS decoder. In addition, computer simulations show that this simple and fast decoding technique reduces the decoding time when compared with existing efficient algorithms including the new Euclidean-algorithm-based decoding approach proposed in this paper.
INTERNATIONAL JOURNAL OF ENGINEERING DEVELOPMENT AND RESEARCH (IJEDR) (ISSN:2321-9939), 2014
Reed-Solomon codes are very useful and effective in burst error in noisy environment. In decoding process for 1 error or 2 errors create easily with using procedure of Peterson-Gorenstein –Zierler algorithm. If decoding process for 3 or more errors, these errors can be solved with key equation of a new algorithm named Berlekamp-massey algorithm. In this paper, wide discussion of procedures of Peterson-Gorenstein –Zierler algorithm and Berlekamp-Massey algorithm and show the advantages of modified version of Berlekam-Massey algorithm with its steps.
A new decoding algorithm for correcting both erasures and errors of reed-solomon codes
IEEE Transactions on Communications, 2003
In this paper, a high efficient decoding algorithm is developed here in order to correct both erasures and errors for Reed-Solomon (RS) codes based on the Euclidean algorithm together with the Berlekamp-Massey (BM) algorithm. The new decoding algorithm computes the errata locator polynomial and the errata evaluator polynomial simultaneously without performing polynomial divisions, and there is no need for the computation of the discrepancies and the field element inversions. Also, the separate computation of the Forney syndrome needed in the decoder is completely avoided. As a consequence, the complexity of this new decoding algorithm is dramatically reduced. Finally, the new algorithm has been verified through a software simulation using C ++ language. An illustrative example of (255,239) RS code using this program shows that the speed of the decoding process is approximately three times faster than that of the inverse-free Berlekamp-Massey algorithm.
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.
Efficient Frequency-Domain Decoding Algorithms for Reed-Solomon Codes
2015
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 .
A Simplified Step-by-Step Decoding Algorithm for Parallel Decoding of Reed-Solomon Codes
IEEE Transactions on Communications, 2007
A simplified parallel step-by-step decoding algorithm is proposed for decoding Reed-Solomon (RS) codes. It uses new method to calculate the determinants of the temporarily changed syndrome matrices, based on the property of these matrices determined in this paper. By using the proposed method, the calculations of the determinants of the temporarily changed syndrome matrices become much simpler and thus the computational complexity of the step-by-step decoding algorithm is significantly reduced.
A Simplified Step-by-Step Decoding Algorithm for Parallel Decoding of Reed–Solomon Codes
IEEE Transactions on Communications, 2000
A simplified parallel step-by-step decoding algorithm is proposed for decoding Reed-Solomon (RS) codes. It uses new method to calculate the determinants of the temporarily changed syndrome matrices, based on the property of these matrices determined in this paper. By using the proposed method, the calculations of the determinants of the temporarily changed syndrome matrices become much simpler and thus the computational complexity of the step-by-step decoding algorithm is significantly reduced.
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.
IEE Proceedings E Computers and Digital Techniques, 1988
Berlekamp 's key equation needed to decode a Reed-Solomon (RS) code. In this article, a simplified procedure is developed and proved. to correct erasures as well as errors by replacing the initial condition of the Euclidean algorithm by the erasure locator polynomial and the Forney syndrome polynomial. By this means, the errata locator polynomial and the errata evaluator polynomial can be obtained, simultaneously and simply, by the Euclidean algorithm only. With this improved technique the complexity of time-domain RS decoders for correcting both errors and erasures is reduced substantially from previous approaches. As a consequence, decoders for correcting both errors and erasures of RS codes can be made more modular, regular, simple, and naturally suitable for both VLSI and software implementation, A n example illustrating this modified decoding procedure is given for a (I 5, 9) RS code. Recently, Eastman' showed that the errata evaluator polynomial can be computed directly by initializing Berlekamp's