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IEEE Information Theory Workshop (ITW)

On the selection of finite alphabet iterative decoders for LDPC codes on the BSC

Ludovic Danjean, David Declercq, Shiva Kumar Planjery, Bane Vasica

Published Oct 1, 2011• 16 citations
Publisher

Abstract

Recently new message passing decoders for LDPC codes, called finite alphabet iterative decoders (FAIDs) were proposed. The messages belong to a finite alphabet and the update functions are simple boolean maps different from the functions used for the belied propagation (BP) decoder. The maps can be chosen using the knowledge of potential trapping sets such that the decoders surpass the BP decoder in the error floor. In this paper, we address the issue of selecting good FAIDs which perform well in the error floor for column weight three codes. We introduce the notion of noisy trapping set which is a generalization based on analyzing the local dynamic behaviour of a given FAID on a trapping set. Using this notion as the core, we provide an iterative greedy algorithm that outputs a set of candidate FAIDs containing potentially good decoders for any given code. To illustrate the appliance of the methodology on several codes, we show that the set of candidate FAIDs contains particularly good FAIDs for different codes with different rates and lengths.

DOI
10.1109/itw.2011.6089476

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Recently new message passing decoders for LDPC codes, called finite alphabet iterative decoders (FAIDs) were proposed. The messages belong to a finite alphabet and the update functions are simple boolean maps different from the functions used for the belied propagation (BP) decoder. The maps can be chosen using the knowledge of potential trapping sets such that the decoders surpass the BP decoder in the error floor. In this paper, we address the issue of selecting good FAIDs which perform well in the error floor for column weight three codes. We introduce the notion of noisy trapping set which is a generalization based on analyzing the local dynamic behaviour of a given FAID on a trapping set. Using this notion as the core, we provide an iterative greedy algorithm that outputs a set of candidate FAIDs containing potentially good decoders for any given code. To illustrate the appliance of the methodology on several codes, we show that the set of candidate FAIDs contains particularly good FAIDs for different 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.
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.
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.
Recently, we introduced a new class of finite alphabet iterative decoders (FAIDs) for low-density parity-check (LDPC) codes. These decoders are capable of surpassing belief propagation (BP) in the error floor region on the binary symmetric channel (BSC) with much lower complexity. In this paper, we introduce a novel scheme with the objective of guaranteeing the correction of a given and potentially large number of errors on column-weight-three LDPC codes. The proposed scheme uses a plurality of FAIDs which collectively correct more error patterns than a single FAID on a given code. The collection of FAIDs utilized by the scheme is judiciously chosen to ensure that individual decoders have different decoding dynamics and correct different error patterns. Consequently, they can collectively correct a diverse set of error patterns, which is referred to as decoder diversity. We provide a systematic method to generate the set of FAIDs for decoder diversity on a given code based on the knowledge of the most harmful trapping sets present in the code. Using the well-known column-weight-three (155,64) Tanner code with dmin= 20 as an example, we describe the method in detail and show, by means of exhaustive simulation, that the guaranteed error correction capability on short length LDPC codes can be significantly increased with decoder diversity.
Finite alphabet iterative decoders (FAID) with multilevel messages that can surpass BP in the error floor region for LDPC codes on the BSC were previously proposed in [1]. In this paper, we propose decimation-enhanced decoders. The technique of decimation which is incorporated into the message update rule, involves fixing certain bits of the code to a particular value. Under appropriately chosen rules, decimation can significantly reduce the number of iterations required to correct a fixed number of errors, while maintaining the good performance of the original decoder in the error floor region. At the same time, the algorithm is much more amenable to analysis. We shall provide a simple decimation scheme for a particularly good 7-level FAID for column-weight three codes on the BSC, that helps to correct a fixed number of errors in fewer iterations, and provide insights into the analysis of the decoder. We shall also examine the conditions under which the decimation-enhanced 7-level FAID performs at least as good as the 7-level FAID.