An application of bivariate polynomial factorization on decoding of Reed-Solomon based codes (original) (raw)

Algebraic list decoding of Reed-Solomon product codes

2006

Product Reed-Solomon codes are widely used in data storage, optical and satellite communication systems. Reed-Solomon product codes can be regarded as evaluation of a bivariate polynomial with constraints on its X and Y -degrees. In this work, we propose polynomial time algorithms to decode Reed-Solomon product codes beyond half the minimum distance. The first algorithm is based on a generalization of the Guruswami-Sudan type decoders. We are able to show that if fraction of number of errors is smaller than 1− 6 p 4R p , where R p is the rate of the product code, then the algorithm can efficiently recover the transmitted codeword. The other algorithm is based on the fact that Reed-Solomon product codes can be viewed as subfieldsubcode of a generalized Reed-Solomon code. So, the decoding algorithms for Reed-Solomon codes are inherited to decoding of RS product codes. By using this fact, we prove that if fraction of number of errors is smaller than 1 − 4 p 4R p then the algorithm is able to recover the transmitted codeword. 1

Novel Polynomial Basis and Its Application to Reed-Solomon Erasure Codes

2014 IEEE 55th Annual Symposium on Foundations of Computer Science, 2014

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.

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.

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.

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 .

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.

Decoding Reed-Solomon codes generated by any generator polynomial

Electronics Letters, 1989

To observe this high degree of conversion, almost perfect phase matching between the two modes is necessary. Theoretically' the conversion efficiency is given by q = (K/?)' sin2 (yL), where L is the waveguide length, K is the coupling constant and y2 = K* + (AB)', AB being the difference between

New Approaches to the Analysis and Design of Reed-Solomon Related Codes

2007

The research that led to this thesis was inspired by Sudan's breakthrough that demonstrated that Reed-Solomon codes can correct more errors than previously thought. This breakthrough can render the current state-of-the-art Reed-Solomon decoders obsolete. Much of the importance of Reed-Solomon codes stems from their ubiquity and utility. This thesis takes a few steps toward a deeper understanding of Reed-Solomon codes as well as toward the design of efficient algorithms for decoding them. After studying the binary images of Reed-Solomon codes, we proceeded to analyze their performance under optimum decoding. Moreover, we investigated the performance of Reed-Solomon codes in network scenarios when the code is shared by many users or applications. We proved that Reed-Solomon codes have many more desirable properties. Algebraic soft decoding of Reed-Solomon codes is a class of algorithms that was stirred by Sudan's breakthrough. We developed a mathematical model for algebraic so...

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.