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IEEE Transactions on Communications

Finite Alphabet Iterative Decoders—Part I: Decoding Beyond Belief Propagation on the Binary Symmetric Channel

Shiva Kumar Planjery, David Declercq, Ludovic Danjean, Bane V. Vasic

Published Sep 16, 2013• 83 citations
Publisher

Abstract

We introduce a new paradigm for finite precision iterative decoding on low-density parity-check codes over the binary symmetric channel. The messages take values from a finite alphabet, and unlike traditional quantized decoders which are quantized versions of the belief propagation (BP) decoder, the proposed finite alphabet iterative decoders (FAIDs) do not propagate quantized probabilities or log-likelihoods and the variable node update functions do not mimic the BP decoder. Rather, the update functions are maps designed using the knowledge of potentially harmful subgraphs that could be present in a given code, thereby rendering these decoders capable of outperforming the BP in the error floor region. On certain column-weight-three codes of practical interest, we show that there exist {FAIDs that surpass the floating-point BP decoder in the error floor region while requiring only three bits of precision for the representation of the messages}. Hence, FAIDs are able to achieve a superior performance at much lower complexity. We also provide a methodology for the selection of FAIDs that is not code-specific, but gives a set of candidate FAIDs containing potentially good decoders in the error floor region for any column-weight-three code. We validate the code generality of our methodology by providing particularly good three-bit precision FAIDs for a variety of codes with different rates and lengths.

DOI
10.1109/tcomm.2013.090513.120443

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We introduce a new paradigm for finite precision iterative decoding on low-density parity-check codes over the binary symmetric channel. The messages take values from a finite alphabet, and unlike traditional quantized decoders which are quantized versions of the belief propagation (BP) decoder, the proposed finite alphabet iterative decoders (FAIDs) do not propagate quantized probabilities or log-likelihoods and the variable node update functions do not mimic the BP decoder. Rather, the update functions are maps designed using the knowledge of potentially harmful subgraphs that could be present in a given code, thereby rendering these decoders capable of outperforming the BP in the error floor region. On certain column-weight-three codes of practical interest, we show that there exist {FAIDs that surpass the floating-point BP decoder in the error floor region while requiring only three bits of precision for the representation of the messages}. Hence, FAIDs are able to achieve a superior performance at much lower complexity. We also provide a methodology for the selection of FAIDs that is not code-specific, but gives a set of candidate FAIDs containing potentially good decoders in the error floor region for any column-weight-three code. We validate the code generality of our methodology by providing particularly good three-bit precision FAIDs for a variety of codes with different rates and lengths.
We introduce a new paradigm for finite precision iterative decoding on low-density parity-check codes over the Binary Symmetric channel. The messages take values from a finite alphabet, and unlike traditional quantized decoders which are quantized versions of the Belief propagation (BP) decoder, the proposed finite alphabet iterative decoders (FAIDs) do not propagate quantized probabilities or log-likelihoods and the variable node update functions do not mimic the BP decoder. Rather, the update functions are maps designed using the knowledge of potentially harmful subgraphs that could be present in a given code, thereby rendering these decoders capable of outperforming the BP in the error floor region. On certain column-weight-three codes of practical interest, we show that there exist 3-bit precision FAIDs that surpass the BP decoder in the error floor. Hence, FAIDs are able to achieve a superior performance at much lower complexity. We also provide a methodology for the selection of FAIDs that is not code-specific, but gives a set of candidate FAIDs containing potentially good decoders in the error floor region for any column-weight-three code. We validate the code generality of our methodology by providing particularly good three-bit precision FAIDs for a variety of codes with different rates and lengths.
The recent renaissance of one particular class of error-correcting codes called low-density parity-check (LDPC) codes has revolutionized the area of communications leading to the so-called field of modern coding theory. At the heart of this theory lies the fact that LDPC codes can be efficiently decoded by an iterative inference algorithm known as belief propagation (BP) which operates on a graphical model of a code. With BP decoding, LDPC codes are able to achieve an exceptionally good error-rate performance as they can asymptotically approach Shannon's capacity. However, LDPC codes under BP decoding suffer from the error floor phenomenon, an abrupt degradation in the error-rate performance of the code in the high signal-to-noise ratio region, which prevents the decoder from achieving very low error-rates. It arises mainly due to the sub-optimality of BP decoding on finite-length loopy graphs. Moreover, the effects of finite precision that stem from hardware realizations of BP decoding can further worsen the error floor phenomenon. Over the past few years, the error floor problem has emerged as one of the most important problems in coding theory with applications now requiring very low error rates and faster processing speeds. Further, addressing the error floor problem while taking finite precision into account in the decoder design has remained a challenge. In this dissertation, we introduce a new paradigm for finite precision iterative decoding of LDPC codes over the binary symmetric channel (BSC). These novel decoders, referred to as finite alphabet iterative decoders (FAIDs), are capable of surpassing the BP in the error floor region at a much lower complexity and memory usage than BP without any compromise in decoding latency. The messages propagated by FAIDs are not quantized probabilities or log-likelihoods, and the variable node update functions do not mimic the BP decoder. Rather, the update functions are simple maps designed to ensure a higher guaranteed error correction capability which improves the error floor performance. We provide a methodology for the design of FAIDs on column-weight-three codes. Using this methodology, we design 3-bit precision FAIDs that can surpass the BP (floating-point) in the error floor region on several column-weight-three codes of practical interest. While the proposed FAIDs are able to outperform the BP decoder with low precision, the analysis of FAIDs still proves to be a difficult issue. Furthermore, their achievable guaranteed error correction capability is still far from what is achievable by the optimal maximum-likelihood (ML) decoding. In order to address these two issues, we propose another novel class of decoders called decimation-enhanced FAIDs for LDPC codes. For this class of decoders, the technique of decimation is incorporated into the variable node update function of FAIDs. Decimation, which involves fixing certain bits of the code to a particular value during decoding, can significantly reduce the number of iterations required to correct a fixed number of errors while maintaining the good performance of a FAID, thereby making such decoders more amenable to analysis. We illustrate this for 3-bit precision FAIDs on column-weight-three codes and provide insights into the analysis of such decoders. We also show how decimation can be used adaptively to further enhance the guaranteed error correction capability of FAIDs that are already good on a given code. The new adaptive decimation scheme proposed has marginally added complexity but can significantly increase the slope of the error floor in the error-rate performance of a particular FAID. On certain high-rate column-weight-three codes of practical interest, we show that adaptive decimation-enhanced FAIDs can achieve a guaranteed error-correction capability that is close to the theoretical limit achieved by ML decoding.
Introduced is a new type of message-passing (MP) decoders for low-density parity-check (LDPC) codes over the binary symmetric channel. Unlike traditional belief propagation (BP) based MP algorithms which propagate probabilities or log-likelihoods, the new MP decoders propagate messages requiring only a finite number of bits for their representation in such a way that good performance in the error floor region is ensured. Additionally, these messages are not quantised probabilities or log-likelihoods. As examples, MP decoders are provided that require only three bits for message representation, but surpass the floating-point BP (which requires a large number of bits for representation) in the error-floor region.
Finite alphabet iterative decoders (FAIDs) for LDPC codes were recently shown to be capable of surpassing the Belief Propagation (BP) decoder in the error floor region on the Binary Symmetric channel (BSC). More recently, the technique of decimation which involves fixing the values of certain bits during decoding, was proposed for FAIDs in order to make them more amenable to analysis while maintaining their good performance. In this paper, we show how decimation can be used adaptively to further enhance the guaranteed error correction capability of FAIDs that are already good on a given code. The new adaptive decimation scheme proposed has marginally added complexity but can significantly improve the slope of the error floor performance of a particular FAID. We describe the adaptive decimation scheme particularly for 7-level FAIDs which propagate only 3-bit messages and provide numerical results for column-weight three codes. Analysis suggests that the failures of the new decoders are linked to stopping sets of the code.
Finite alphabet iterative decoders (FAIDs) for LDPC codes were recently shown to be capable of surpassing the Belief Propagation (BP) decoder in the error floor region on the Binary Symmetric channel (BSC). More recently, the technique of decimation which involves fixing the values of certain bits during decoding, was proposed for FAIDs in order to make them more amenable to analysis while maintaining their good performance. In this paper, we show how decimation can be used adaptively to further enhance the guaranteed error correction capability of FAIDs that are already good on a given code. The new adaptive decimation scheme proposed has marginally added complexity but can significantly improve the slope of the error floor performance of a particular FAID. We describe the adaptive decimation scheme particularly for 7-level FAIDs which propagate only 3-bit messages and provide numerical results for column-weight three codes. Analysis suggests that the failures of the new decoders are linked to stopping sets of the code.
In this paper, we propose a new approach to design finite alphabet iterative decoders (FAIDs) for Low-Density Parity Check (LDPC) codes over binary symmetric channel (BSC) via recurrent quantized neural networks (RQNN). We focus on the linear FAID class and use RQNNs to optimize the message update look-up tables by jointly training their message levels and RQNN parameters. Existing neural networks for channel coding work well over Additive White Gaussian Noise Channel (AWGNC) but are inefficient over BSC due to the finite channel values of BSC fed into neural networks. We propose the bit error rate (BER) as the loss function to train the RQNNs over BSC. The low precision activations in the RQNN and quantization in the BER cause a critical issue that their gradients vanish almost everywhere, making it difficult to use classical backward propagation. We leverage straight-through estimators as surrogate gradients to tackle this issue and provide a joint training scheme. We show that the framework is flexible for various code lengths and column weights. Specifically, in high column weight case, it automatically designs low precision linear FAIDs with superior performance, lower complexity, and faster convergence than the floating-point belief propagation algorithms in waterfall region.