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IEEE Vehicular Technology Conference

A proof of the hadamard transform decoding of the belief propagation algorithm for LDPCC over GF(q)

Xiangming Li, Mohammad Soleymani

Published Apr 12, 2005• 17 citations
Publisher

Abstract

The LDPCC (low-density parity-check codes) belief propagation decoding algorithm uses a bipartite graph and sends belief messages between the variable symbols and the checks in an iterative process. The message that a check sends to a variable symbol can be calculated using a Hadamard transform, as introduced by T.J. Richardson and R.L. Urbanke (see IEEE Trans. Inform. Theory, vol.IT-47, p.599-618, 2001). However, the explicit proof of the correctness of the FHT (fast Hadamard transform) implementation of the decoding algorithm has not, so far, been seen in the literature. We give a proof for the FHT implementation of the decoding algorithm for LDPC codes over GF(q).

DOI
10.1109/vetecf.2004.1400507

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The LDPCC (low-density parity-check codes) belief propagation decoding algorithm uses a bipartite graph and sends belief messages between the variable symbols and the checks in an iterative process. The message that a check sends to a variable symbol can be calculated using a Hadamard transform, as introduced by T.J. Richardson and R.L. Urbanke (see IEEE Trans. Inform. Theory, vol.IT-47, p.599-618, 2001). However, the explicit proof of the correctness of the FHT (fast Hadamard transform) implementation of the decoding algorithm has not, so far, been seen in the literature. We give a proof for the FHT implementation of the decoding algorithm for LDPC codes over GF(q).
Low-density parity-check (LDPC) codes over finite fields GF(q) have a better performance than those of the binary low-density parity-check codes, at short and medium block lengths, but the decoder of GF(q)-LDPC has more complexity. In this paper, we present a new reduced-complexity belief propagation (BP) decoding algorithm for GF(q)-LDPC codes by analyzing the reliability of code symbols. Instead of update all the variable nodes during iteration, we just update variable nodes which are mostly to be in an error. The algorithm also shows a property to reduce the effects of oscillating on decoding. The simulations prove that we could improve performance and reduce complexity at the same time for short and medium GF(q)-LDPC codes.
A message-passing decoder for GF(q) low-density parity-check codes is defined, which uses discrete messages from a subset of all possible binary vectors of length q. The proposed algorithm is a generalization to GF(q) of Richardson and Urbanke's decoding "Algorithm E" for binary codes. Density evolution requires a mapping between the probability distribution spaces for the channel, variable and check messages, and under the proposed algorithm, exact density evolution is possible. Symmetries in the message densities permit reduction in the size of the probability distribution space. Noise thresholds are obtained for LDPC codes on discrete memoryless channels, and as with Algorithm E, are remarkably close to noise thresholds under more complex belief propagation decoding.
Variant belief propagation (BP) algorithms are applied to low-density parity-check (LDPC) codes. However, conventional decoders suffer from a large resource consumption due to gathering messages from all the neighbour variable-nodes and/or check-nodes through cumulative calculations. In this paper, a check-belief propagation (CBP) decoding algorithm is proposed. Check-belief is used as the probability that the corresponding parity-check is satisfied. All check-beliefs are iteratively enlarged in a sequential recursive order, and successful decoding will be achieved after the check-beliefs are all big enough. Compared to previous algorithms employing a large number of cumulative calculations to gather all the neighbour messages, CBP decoding can renew each check-belief by propagating it from one check-node to another through only one variable-node, resulting in a low complexity decoding with no cumulative calculations. The simulation results and analyses show that the CBP algorithm provides little error-rate performance loss in contrast with the previous BP algorithms, but consumes much fewer calculations and memories than them. It earns a big benefit in terms of complexity.
We propose an augmented belief propagation (BP) decoder for low-density parity check (LDPC) codes which can be utilized on memoryless or intersymbol interference channels. The proposed method is a heuristic algorithm that eliminates a large number of pseudocodewords that can cause nonconvergence in the BP decoder. The augmented decoder is a multistage iterative decoder, where, at each stage, the original channel messages on select symbol nodes are replaced by saturated messages. The key element of the proposed method is the symbol selection process, which is based on the appropriately defined subgraphs of the code graph and/or the reliability of the information received from the channel. We demonstrate by examples that this decoder can be implemented to achieve substantial gains (compared to the standard locally-operating BP decoder) for short LDPC codes decoded on both memoryless and intersymbol interference Gaussian channels. Using the Margulis code example, we also show that the augmented decoder reduces the error floors. Finally, we discuss types of BP decoding errors and relate them to the augmented BP decoder.
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.