Multilevel Decoders Surpassing Belief Propagation on the Binary Symmetric Channel

In this paper, we propose a new class of quantized message-passing decoders for LDPC codes over the BSC. The messages take values (or levels) from a finite set. The update rules do not mimic belief propagation but instead are derived using the knowledge of trapping sets. We show that the update rules can be derived to correct certain error patterns that are uncorrectable by algorithms such as BP and min-sum. In some cases even with a small message set, these decoders can guarantee correction of a higher number of errors than BP and min-sum. We provide particularly good 3-bit decoders for 3-left-regular LDPC codes. They significantly outperform the BP and min-sum decoders, but more importantly, they achieve this at only a fraction of the complexity of the BP and min-sum decoders.Comment: 5 pages, in Proc. of 2010 IEEE International Symposium on Information Theory (ISIT

Analysis and Design of Binary Message-Passing Decoders

Binary message-passing decoders for low-density parity-check (LDPC) codes are studied by using extrinsic information transfer (EXIT) charts. The channel delivers hard or soft decisions and the variable node decoder performs all computations in the L-value domain. A hard decision channel results in the well-know Gallager B algorithm, and increasing the output alphabet from hard decisions to two bits yields a gain of more than 1.0 dB in the required signal to noise ratio when using optimized codes. The code optimization requires adapting the mixing property of EXIT functions to the case of binary message-passing decoders. Finally, it is shown that errors on cycles consisting only of degree two and three variable nodes cannot be corrected and a necessary and sufficient condition for the existence of a cycle-free subgraph is derived.Comment: 8 pages, 6 figures, submitted to the IEEE Transactions on Communication

The Effect of Saturation on Belief Propagation Decoding of LDPC Codes

We consider the effect of LLR saturation on belief propagation decoding of low-density parity-check codes. Saturation occurs universally in practice and is known to have a significant effect on error floor performance. Our focus is on threshold analysis and stability of density evolution. We analyze the decoder for certain low-density parity-check code ensembles and show that belief propagation decoding generally degrades gracefully with saturation. Stability of density evolution is, on the other hand, rather strongly affected by saturation and the asymptotic qualitative effect of saturation is similar to reduction of variable node degree by one.Comment: Submitted to ISIT. Longer version to be submitted to IT Transactions in preparatio

Density Evolution and Functional Threshold for the Noisy Min-Sum Decoder

This paper investigates the behavior of the Min-Sum decoder running on noisy devices. The aim is to evaluate the robustness of the decoder in the presence of computation noise, e.g. due to faulty logic in the processing units, which represents a new source of errors that may occur during the decoding process. To this end, we first introduce probabilistic models for the arithmetic and logic units of the the finite-precision Min-Sum decoder, and then carry out the density evolution analysis of the noisy Min-Sum decoder. We show that in some particular cases, the noise introduced by the device can help the Min-Sum decoder to escape from fixed points attractors, and may actually result in an increased correction capacity with respect to the noiseless decoder. We also reveal the existence of a specific threshold phenomenon, referred to as functional threshold. The behavior of the noisy decoder is demonstrated in the asymptotic limit of the code-length -- by using "noisy" density evolution equations -- and it is also verified in the finite-length case by Monte-Carlo simulation.Comment: 46 pages (draft version); extended version of the paper with same title, submitted to IEEE Transactions on Communication

Mutual Information-Maximizing Quantized Belief Propagation Decoding of Regular LDPC Codes

In mutual information-maximizing lookup table (MIM-LUT) decoding of low-density parity-check (LDPC) codes, table lookup operations are used to replace arithmetic operations. In practice, large tables need to be decomposed into small tables to save the memory consumption, at the cost of degraded error performance. In this paper, we propose a method, called mutual information-maximizing quantized belief propagation (MIM-QBP) decoding, to remove the lookup tables used for MIM-LUT decoding. Our method leads to a very efficient decoder, namely the MIM-QBP decoder, which can be implemented based only on simple mappings and fixed-point additions. Simulation results show that the MIM-QBP decoder can always considerably outperform the state-of-the-art MIM-LUT decoder, mainly because it can avoid the performance loss due to table decomposition. Furthermore, the MIM-QBP decoder with only 3 bits per message can outperform the floating-point belief propagation (BP) decoder at high signal-to-noise ratio (SNR) regions when testing on high-rate codes with a maximum of 10-30 iterations