Performance of Reed–Solomon codes using the Guruswami–Sudan algorithm with improved interpolation efficiency

Abstract

List decoding is a novel method for decoding Reed-Solomon (RS) codes that generates a list of candidate transmitted messages instead of one unique message as with conventional algebraic decoding, making it possible to correct more errors. The Guruswami-Sudan (GS) algorithm is the most efficient list decoding algorithm for RS codes. Until recently only a few papers in the literature suggested practical methods to implement the key steps (interpolation and factorisation) of the GS algorithm that make the list decoding of RS codes feasible. However, the algorithm's high decoding complexity is unsolved and a novel complexity-reduced modification to improve its efficiency is presented. A detailed explanation of the GS algorithm with the complexity-reduced modification is given with simulation results of RS codes for different list decoding parameters over the AWGN and Rayleigh fading channels. A complexity analysis is presented comparing the GS algorithm with our modified GS algorithm, showing the modification can reduce complexity significantly in low error weight situations. Simulation results using the modified GS algorithm show larger coding gains for RS codes with lower code rates, with more significant gains being achieved over the Rayleigh fading channels.

A new progressive algebraic soft decoding algorithm for reed-solomon codes

“…In decoding an (n, k) RS code, monomials x a y b are organized by the (1, k − 1)-revlex order [5]. Given a polynomial…”

“…In [5], interpolation complexity is reduced by eliminating the polynomials with a leading order greater than the number of interpolation constraints. The interpolation complexity can also be reduced by utilizing the unique decoding outcome [6], which leads to a reduction of the interpolation multiplicity.…”

List Decoding

“…Therefore, reducing the complexity of interpolation is essential to improving the algorithm's efficiency. The leading order of the polynomial group G i k is defined as the minimal leading order (lod) among the group's polynomials [13]:…”

Efficient Factorisation Algorithm for List Decoding Algebraic-Geometric and Reed-Solomon Codes

“…At the end, the minimal weight polynomial in the list is chosen as Q(z). A detailed tutorial discussion with a complexity reduced scheme of this algorithm is given in [9]. Factorisation finds the z roots of Q(z).…”

“…By redefining a polynomial over a Hermitian function field, Hoholdt and Nielsen [8] extended Koetter's algorithm to list decode Hermitian codes. In order to improve interpolation efficiency, the authors recently proposed a complexity reduced scheme [9] which can be applied to both RS and AG codes. To implement factorisation, Gao and Shokrollahi [10] proposed an algorithm to find roots of polynomials defined over function fields of plane curves.…”