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IEEE AFRICON

Belief propagation for nonlinear block codes

Etienne Ackermann, Trienko Grobler, Gusti van Zyl, Jan Corne Olivier

Published Sep 1, 2011• 0 citations
Publisher

Abstract

Special cases of the belief propagation algorithm are commonly used to decode various linear error correcting codes. We show that the belief propagation algorithm can also be applied to the decoding problem for nonlinear error correcting codes by considering a modified set of update rules. We demonstrate the modified belief propagation algorithm by applying it to the well-known Nordstrom-Robinson (15, 256, 5) nonlinear code. The Nordstrom-Robinson code contains many cycles, which means that belief propagation is suboptimal. Nevertheless, we observe a coding gain of about 1.2 dB with this method.

DOI
10.1109/afrcon.2011.6072001

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Special cases of the belief propagation algorithm are commonly used to decode various linear error correcting codes. We show that the belief propagation algorithm can also be applied to the decoding problem for nonlinear error correcting codes by considering a modified set of update rules. We demonstrate the modified belief propagation algorithm by applying it to the well-known Nordstrom-Robinson (15, 256, 5) nonlinear code. The Nordstrom-Robinson code contains many cycles, which means that belief propagation is suboptimal. Nevertheless, we observe a coding gain of about 1.2 dB with this method.
Weighted belief propagation (WBP) for the decoding of linear block codes is considered. In WBP, the Tanner graph of the code is unrolled with respect to the iterations of the belief propagation decoder. Then, weights are assigned to the edges of the resulting recurrent network and optimized offline using a training dataset. The main contribution of this paper is an adaptive WBP where the weights of the decoder are determined for each received word. Two variants of this decoder are investigated. In the parallel WBP decoders, the weights take values in a discrete set. A number of WBP decoders are run in parallel to search for the best sequence- of weights in real time. In the two-stage decoder, a small neural network is used to dynamically determine the weights of the WBP decoder for each received word. The proposed adaptive decoders demonstrate significant improvements over the static counterparts in two applications. In the first application, Bose-Chaudhuri-Hocquenghem, polar and quasi-cyclic low-density parity-check (QC-LDPC) codes are used over an additive white Gaussian noise channel. The results indicate that the adaptive WBP achieves bit error rates (BERs) up to an order of magnitude less than the BERs of the static WBP at about the same decoding complexity, depending on the code, its rate, and the signal-to-noise ratio. The second application is a concatenated code designed for a long-haul nonlinear optical fiber channel where the inner code is a QC-LDPC code and the outer code is a spatially coupled LDPC code. In this case, the inner code is decoded using an adaptive WBP, while the outer code is decoded using the sliding window decoder and static belief propagation. The results show that the adaptive WBP provides a coding gain of 0.8 dB compared to the neural normalized min-sum decoder, with about the same computational complexity and decoding latency.
A novel belief propagation(BP) decoding algorithm for convolutional low-density parity-check codes is proposed.The proposed algorithm raises the efficiency of updating the variable information by applying feedback information at each decoding iteration and employs the weighting factor to reduce the error propagation caused by the cycles in the Tanner graph,thus yielding a faster convergence of the decoding.Simulation results show that an error performance better than that of the existing belief propagation algorithm can be achieved,while the 5/8 decoding delay and the computation complexity are effectively reduced.Compared with the existing BP,the proposed algorithm achieves a gain of 0.16dB at the BER of 10-6 with the same number of iterations.
Abstract — The exact nonlinear loop gain of the belief propagation algorithm (BPA) in its log-likelihood ratio (LLR) formulation is computed. The nonlinear gains for regular lowdensity parity-check (LDPC) error correcting codes can be computed exactly using a simple formula. It is shown that in some neighborhood of the origin this gain is actually much smaller than the identity. Using a small-gain argument, this implies that the BPA is in fact locally input-to-state stable and produces bounded outputs for small-in-norm input LLR vectors. In a larger domain the algorithm produces at least bounded trajectories. Further it is shown that, as the block length increases, these regions exponentially shrink. Index Terms — iterative decoding; LDPC codes; dynamical system; convergence; belief propagation; small-gain theorem I.
In this paper, two iterative message passing algorithms based on belief propagation proposed for the decoding of any binary linear block code are compared. The innovation of these algorithms lies in the fact that the structure of the graph of the parity check matrix is adaptively modified to render it suitable to belief propagation decoding. The modification is done such that the least reliable bits received from the channel are leaves of the graph. These algorithms perform favorably when compared with existing hard and iterative soft decision decoding algorithms in terms of error rate while maintaining a complexity polynomial in block length. It was found that while these two algorithms are conceptually similar, their performance in terms of word error rate (as a function of signal-to-noise ratio), decoding time (average number of iterations) and the updated bit reliabilities are very different
Abstract In this letter, a modified message passing algorithm to prevent short cycles in the factor graph is proposed and combined with a relaxation technique to improve the performance of cyclic redundancy check aided belief propagation decoding for polar codes. Numerical results show that the proposed scheme can achieve a 0.7 dB coding gain, for the code rate of 0.5 at the block error rate of 10 −5 with a comparable average number of iterations over the pure belief propagation decoding scheme when the code block length is 1024.
Short block-length polar-like codes showcase exceptional error correction performance (ECP) using sequential decoding or successive cancellation list decoding with a large list size. However, achieving a high level of reliability with these methods involves high-latency decoding. To meet the growing demand for low-latency communication with acceptable complexity, belief propagation (BP) decoding emerges as an attractive option, although its ECP is known to fall short of those high-latency alternatives. In this paper, we propose an enhanced BP decoding approach for polar codes, leveraging reinforcement learning (RL) to optimize the message-passing schedule. Moreover, we investigate the design of the polar code rate profile and corresponding Tanner graph representation to enhance the benefits of RL. Numerical results demonstrate a performance gain of more than 1 dB for polar codes with a length of 128 and a rate of 0.5 compared to conventional BP decoding alone at highEb/N0values, demonstrating the promise of the proposed approach.