A Multiple-Voting-Based Decoding Algorithm for Nonbinary LDPC-Coded Modulation Systems


We propose a multiple-voting-based joint detection-decoding algorithm for nonbinary lowdensity parity-check (LDPC)-coded modulation systems. This algorithm is inspired from the reliabilitybased JDD algorithm for nonbinary LDPC-coded modulation systems, that has been proposed recently, in which the accumulated reliability of symbols based on one-step majority-logic decoding algorithm and the Chase-like local list decoding algorithm are used. However, the reliability-based JDD algorithm still has a significant performance degradation of at least 1 dB with low column weight (dv ≤ 4). In order to reduce the performance degradation with low column weight, the proposed algorithm allows unfixed number of variable nodes to pass two symbols to the associated check node, in contrast with the reliability-based JDD algorithm, which allows only one variable node to pass two symbols to check node, when updating variable-to-check messages. Moreover, the votes are weighted differently according to the components of the list in the checksum computation. Simulations show that the proposed algorithm yields better performance with low column weight, while still maintaining the low complexity feature. INTRODUCTION

With the invention and rediscovery of binary low-density parity-check (LDPC) codes [1]–[3], many works about the construction, decoding, performance analysis and applications for binary LDPC codes have been done intensively. Binary LDPC codes have been shown to approach the Shannon limit performance using long codewords. Later, Davey and MacKay put forward to a class of nonbinary LDPC codes defined over the finite field Fq [4], referred to as q-ary LDPC codes here. They also introduced a sum-product algorithm for decoding q-ary LDPC codes, which is known as QSPA. It provides the best error performance at the cost of the highest computational complexity. Significant works on the design, construction and analysis of nonbinary LDPC codes have been studied since the middle of 2000. Results in these works show that nonbinary LDPC codes have great potential for short and moderate code lengths compared with their binary counterparts.

However, the gains that nonbinary LDPC codes provides have to be balanced by the increased decoding complexity of nonbinary LDPC decoders, which prevents the implementation of nonbinary LDPC decoders in practical systems. In order to reduce the decoding complexity, the fast Fourier transform based QSPA (FFT-QSPA) was proposed in [5]. The complexity of the FFT-based SPA is still too high. Hence, several new approaches, such as extended minsum (EMS) algorithm presented by Declercq and Fossorier in [6] and stochastic decoding [7], have been proposed to reduce the hardware implementation complexity. Moreover, several improved algorithms based on EMS algorithm were presented recently [8]–[10]. These soft-decision algorithms improve the computations of configuration sets for check nodes, but still have higher decoding complexity for highrate nonbinary LDPC codes, where the degree of check nodes takes large values. Several alternatives are also presented to reduce the decoding complexity. One is the hard-decision algorithm including the one-step majority-logic (OSMLG) decoding algorithm and the bit-flipping (BF) decoding algorithm [11], which has very low decoding complexity. However, they only apply to nonbinary LDPC codes with high column weight, and suffer a large performance loss. Another is the reliability-based decoding algorithm which gives a compromise between performance and decoding complexity to some extent. Lin et al. [12], [13] proposed two reliability-based massage passing algorithms (the iterative hard- and soft- reliability based majority logic algorithms) for decoding nonbinary LDPC codes. Huang et al. [14] presented bit-reliability based majority logic decoding algorithms for nonbinary LDPC codes. They achieve good tradeoff between performance and decoding complexity. Besides, in [15] and [16], symbol-reliability based decoding algorithms were also proposed, which suffers the performance loss when the nonbinary LDPC codes have low column weight. In [17] and [18], a symbol flipping decoder based on multiple votes was


In this paper, we introduced a multiple-voting based joint detection-decoding algorithm for nonbinary LDPC-coded modulation systems, aiming at reducing the performance degradation with low column weight (dv ≤ 4) compared to the FFT-QSPA. Simulation results show that, with the column weights 4, 3, and 2, the multiple-voting based joint detection-decoding algorithm is about 0.5 dB, 0.75 dB, and 1 dB away from the FFT-QSPA, respectively. Besides, the multiple-voting based IJDD algorithm outperforms the reliability-based JDD algorithm in [21] by about 0.7 dB, 0.2 dB, and 0.3 dB at a BER of 10−5 with the column weights 4, 3, and 2, respectively. Furthermore, the complexity comparison of the FFT-QSPA, the reliability-based JDD and the multiple-voting based IJDD algorithms was given. Therefore, it can be concluded that the multiple-voting based joint detection-decoding algorithm provides better trade-off between the performance and complexity for the nonbinary LDPC codes with low-column weight.