Abstract
This work presents a modified extended key equation algorithm in list decoding of generalized Reed-Solomon (GRS) codes. A list decoding algorithm of generalized Reed-Solomon codes has two steps, interpolation and factorization. The extended key equation algorithm (EKE) is an interpolation-based approach with a lower complexity than Sudan's algorithm. To increase the decoding speed, this work proposes a modified EKE algorithm to perform codeword checking prior to such an interpolation process. Since the evaluation mapping is engaged in encoding, a codeword is not generated systematically. Thus, the transmission information is not directly obtained from a received codeword. Therefore, the proposed algorithm undertakes a matrix operation to obtain the transmission information once a received vector has been checked to be error-free. Simulation results demonstrate that the modified EKE algorithm in list decoding of a GRS code provides low complexity, particularly at high signal-to-noise ratios.
, producing a nonzero bivariate polynomial 
with at least 
points 
, such that 
and 
. Factorization is then performed on 
, yielding linear factors (or called 
-root polynomials) 
. The codewords are then generated from these distinct factors 
via an evaluation mapping. A decoded codeword 
is chosen if the Hamming distance between 
and 
is 
or less.
, an EKE algorithm has been presented to decrease this complexity [
. 
represents a design parameter, typically a small constant, which is an upper bound on the size of the list of decoded codewords.
at each interpolation point. A nonzero 
polynomial exists that interpolates the points 
, 
with multiplicity 
, and is formed by 
, where 
, and the expression of 
denotes the number of ways to choose 
from 
. In comparison with Sudan's work, the GS algorithm provides more 
linear homogeneous equations in interpolation, thus improving the decoding correction distance. Increasing 
improves the decoding performance but also increases the required complexity. The asymptotical decoding correction fraction is given by 
, and the code rate 
is given by 
. The increase in decoding capability is substantial, especially for low-rate GRS codes.
, the KV algorithm can achieve a coding gain of about 1 dB over the GS algorithm, for a (255, 144) GRS code transmitted over an additive white Gaussian noise (AWGN) channel using 256-QAM modulation [
and 
. A codeword in an 
GRS code over GF(
) is generated as
, the generator matrix is
is a primitive element in GF(
).
is the upper bound of the number of consistent codewords, which are at Hamming distance 
from any received word. For an 
GRS code, Sudan's algorithm corrects any error pattern of up to 
errors for
denotes the smallest nonnegative integer holding the following equation:
, the value of 
becomes
be a field, and let 
represent the set of all polynomials of degree 
in the variable 
over 
. Sudan's algorithm consists of the following steps.
over 
with at least 
points 
, such that 
and 
, for a received vector 
.
for which 
is a factor of 
and 
for at least 
locators 
.
RS code, the error-locator polynomial 
and the error-evaluator polynomial 
are computed in the following key equation (KE):
where the polynomial arithmetic is carried out modulo a power of 
in Sudan's algorithm, an EKE algorithm is derived as follows:
, 
, and 
are polynomials that satisfy certain degree constraints and 
are syndrome polynomials computed as follows:
and 
. After these polynomials 
, 
, have been computed in (11) by using the Feng-Tzeng algorithms [
is obtained as follows:
in [
, from (4), the value of 
is determined by the following range:
-matrix in (11) are 
by 
. Since the Feng-Tzeng algorithm is run column by column in a matrix, therefore the column length dominates the decoding complexity. Reducing the column length lowers the complexity of locating the smallest set of linear dependent coefficients. The algorithm of [
homogeneous linear equations in (11) and then finding the corresponding coefficients of 
in (14). Hence, the time complexity is 
, which is less than the time complexity of 
of Sudan's algorithm. Consequently, the EKE algorithm is more attractive than the algorithm of [
is associated with a codeword 
, which has zeros 
[
is given by [
, codeword checking 
is equal to the computation of 
, which is
denotes the matrix transpose.
in (9), a different element 
should yield a different result for 
. Consequently, the value of 
can be simplified as
, the calculation of each 
can be denoted as follows:
becomes
is checked to be all-zero, and then the transmission message is given by
is the last 
-tuple of 
and 
is a 
matrix such that 
is a systematical matrix.
, a codeword 
is given by the evaluation mapping provided as the transmission message 
. If the codeword is transmitted over an error-free channel, then 
is checked to be all-zero by (17) and the transmission message is computed by (22) as follows:
, 
GRS code is adjusted as follows. 
, for a received vector 
. If 
is an all-zero vector, then output the corresponding message vector 
determined as 
, where a vector 
is the last 
-tuple of 
. Then, go to step 5.
and 
and 
,
, where 
, by the Feng-Tzeng algorithm such that 
, 
,
, and then obtain the polynomial 
satisfying 
, 
.
by employing the reconstruction algorithm [
-root polynomials 
.
for each polynomial 
. Output the message vectors 
of the codewords 
with the Hamming distance to 
equal to 
or less.
, the probabilities of the EKE algorithm being utilized in list decoding of these two GRS codes are 0.98 and 0.75, respectively. This work presents a modified EKE algorithm, incorporating codeword checking and a matrix operation, which obtains the transmission information from the received codeword. The computation of codeword checking does not increase the complexity of the original EKE algorithm, because it is an item in the original decoding process. The proposed EKE algorithm is beneficial when the signal-to-noise ratio is high.
