A fast Generalized Minimum Distance decoder for Reed-Solomon codes based on the extended Euclidean algorithm

Abstract

This paper presents a method to determine a set of basis polynomials from the extended Euclidean algorithm that allows Generalized Minimum Distance decoding of ReedSolomon codes with a complexity of O(nd).

Adaptive single-trial error/erasure decoding of Reed-Solomon codes

“…without increasing complexity. A similar result was achieved in [4], [5] for the Euclidean algorithm and in [6] using Newton interpolation. Up to our knowledge, there is no such modification of Guruswami-Sudan (GS) [7] list decoding -which can be considered as state of the art in algebraic decoding -so far.…”

“…,h n−1 ),h 0 ≥ · · · ≥h n−1 , by averaging over 10 4 random unreliability vectors. For each variant of P (τ ) (exact, Hoeffding approximation, ε 0 (τ ) approximation), we calculateτ := τ ⋆ forh according to (4). We useτ for every received vector, which means that the simulation is in fact non-adaptive, using the optimal erasing strategy for the average unreliability vector.…”

The Euclidean algorithm for Generalized Minimum Distance decoding of Reed-Solomon codes

“…which has a form similar to the key equation (5). This implies that the EEA can be used for solving (5). Hence by setting…”

“…In [5] it was shown that c (j+1) , the number of coefficients of q (j+1) (x) that can be determined from the syndrome, is given by…”

“…in 1975 [3], and is based on the extended Euclidean algorithm (EEA). Recently, there have been attempts to perform GMD decoding, also with a complexity of O(n 2 ), using polynomials that are obtained from the extended Euclidean algorithm [4], [5]. Since the BMA and EEA are known to be equivalent in the decoding of RS codes up to half the minimum distance, it is interesting to try to find an extension to the EEA that is equivalent to Kötters extension to the BMA.…”