Efficient decoding of Reed-Solomon codes beyond half the minimum distance

Abstract

A list decoding algorithm is presented for [ ] Reed-Solomon (RS) codes over GF ( ), which is capable of correcting more than ( ) 2 errors. Based on a previous work of Sudan, an extended key equation (EKE) is derived for RS codes, which reduces to the classical key equation when the number of errors is limited to ( ) 2 . Generalizing Massey's algorithm that finds the shortest recurrence that generates a given sequence, an algorithm is obtained for solving the EKE in time complexity ( ( ) ), where is a design parameter, typically a small constant, which is an upper bound on the size of the list of decoded codewords. (The case = 1 corresponds to classical decoding of up to ( ) 2 errors where the decoding ends with at most one codeword.) This improves on the time complexity ( ) needed for solving the equations of Sudan's algorithm by a naive Gaussian elimination. The polynomials found by solving the EKE are then used for reconstructing the codewords in time complexity (( log ) ( + log )) using root-finders of degree-univariate polynomials.

QR code image detection using run-length coding

“…To improve the ability of recognition, we use Reed-Solomon (RS) correcting algorithm to generate RS code so as correct the data of QR code. Reed-Solomon code is a kind of extended non-binary BCH code calculated in galois field [6]. We put the RS code behind the data stream and transmit them together.…”

Factorization-free Low-complexity Chase soft-decision decoding of Reed-Solomon codes

“…In the re-encoded decoder, the coefficients of γ(x) can be computed iteratively by using the Roth-Ruckenstein factorization algorithm [11]. This algorithm computes one coefficient of γ(x) in each iteration, and each iteration consists of two steps: root computation and polynomial updating.…”

FPGA implementation of a factorization processor for soft-decision reed-solomon decoding

“…Assuming the bivariate polynomial output from the interpolation step is Q(X, Y ), the factorization determines all the factors of Q(X, Y ) in the form of Y − f (X) with deg(f (X)) < k. The algorithm in [5] can be described by the pseudo-code below.…”

“…We will focus on the factorization step in this paper. The algorithm proposed by Roth and Ruckenstein [5] is the most efficient to solve the factorization problem. Based on this algorithm, several factorization architectures [6] [7][8] [9] were proposed for the hardware implementation.…”