Shiva Kumar Planjery

Trapping set ontology

The failures of iterative decoders for low-density parity-check (LDPC) codes on the additive white Gaussian noise channel (AWGNC) and the binary symmetric channel (BSC) can be understood in terms of combinatorial objects known as trapping sets. In this paper, we derive a systematic method to identify the most relevant trapping sets for decoding over the BSC in the error floor region. We elaborate on the notion of the critical number of a trapping set and derive a classification of trapping sets. We then develop the trapping set ontology, a database of trapping sets that summarizes the topological relations among trapping sets. We elucidate the usefulness of the trapping set ontology in predicting the error floor as well as in designing better codes.

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.

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

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.

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

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.

Iterative decoding beyond belief propagation

At the heart of modern coding theory lies the fact that low-density parity-check (LDPC) codes can be efficiently decoded by belief propagation (BP). The BP is an inference algorithm which operates on a graphical model of a code, and lends itself to low-complexity and high-speed implementations, making it the algorithm of choice in many applications. It has unprecedentedly good error rate performance, so good that when decoded by the BP, LDPC codes approach theoretical limits of channel capacity. However, this capacity approaching property holds only in the asymptotic limit of code length, while codes of practical lengths suffer abrupt performance degradation in the low noise regime known as the error floor phenomenon. Our study of error floor has led to an interesting and surprising finding that it is possible to design iterative decoders which are much simpler yet better than belief propagation! These decoders do not propagate beliefs but a rather different kind of messages that reflect the local structure of the code graph. This has opened a plethora of exciting theoretical problems and applications. This paper introduces this new paradigm.

Finite alphabet iterative decoding (FAID) of the (155,64,20) Tanner code

It is now well established that iterative decoding approaches the performance of Maximum Likelihood Decoding of sparse graph codes, asymptotically in the block length. For a finite length sparse code, iterative decoding fails on specific subgraphs generically termed as trapping sets. Trapping sets give rise to error floor, an abrupt degradation of the code error performance in the high signal to noise ratio regime. In this paper, we will study a recently introduced class of quantized iterative decoders, for which the messages are defined on a finite alphabet and which successfully decode errors on subgraphs that are uncorrectable by conventional decoders such as the min-sum or the belief propagation. We will especially study the performance of the proposed finite alphabet iterative decoders on the famous (155,64,20) Tanner code.

Finite Alphabet Iterative Decoders—Part II: Towards Guaranteed Error Correction of LDPC Codes via Iterative Decoder Diversity

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 approaching maximum likelihood performance on the Binary Symmetric Channel

We introduce a generic approach for improving the guaranteed error correction capability of regular low-density parity check codes. The method relies on operating (in serial or in parallel) a set of finite alphabet iterative decoders. The message passing update rules are judiciously chosen to ensure that decoders have different dynamics on a specific finite-length code. The idea is that for the Binary Symmetric Channel, if some error pattern cannot be corrected by one particular decoder, there exists in the set of decoders, another decoder which can correct this pattern. We show how to select a plurality of message update rules so that the set of decoders can collectively correct error patterns on the dominant trapping sets. We also show that a set of decoders with dynamic re-initializations can approach the performance of maximum likelihood decoding for finite-length regular column-weight three codes.

Finite Alphabet Iterative Decoders for LDPC Codes: Optimization, Architecture and Analysis

Low-density parity-check (LDPC) codes are adopted in many applications due to their Shannon-limit approaching error-correcting performance. Nevertheless, belief-propagation (BP) based decoding of these codes suffers from the error-floor problem, i.e., an abrupt change in the slope of the error-rate curve that occurs at very low error rates. Recently, a new type of decoders termed finite alphabet iterative decoders (FAIDs) were introduced. The FAIDs use simple Boolean maps for variable node processing, and can surpass the BP-based decoders in the error floor region with very short word length. We restrict the scope of this paper to regular dv=3 LDPC codes on the BSC channel. This paper develops a low-complexity implementation architecture for the FAIDs by making use of their properties. Particularly, an innovative bit-serial check node unit is designed for the FAIDs, and a small-area variable node unit is proposed by exploiting the symmetry in the Boolean maps. Moreover, an optimized data scheduling scheme is proposed to increase the hardware utilization efficiency. From synthesis results, the proposed FAID implementation needs only 52% area to reach the same throughput as one of the most efficient standard Min-Sum decoders for an example (7807, 7177) LDPC code, while achieving better error-correcting performance in the error-floor region. Compared to an offset Min-Sum decoder with longer word length, the proposed design can achieve higher throughput with 45% area, and still leads to possible performance improvement in the error-floor region.

Approaching maximum likelihood decoding of finite length LDPC codes via FAID diversity

We introduce a generic approach, called FAID diversity, for improving the error correction capability of regular low-density parity check codes, beyond the belief propagation performance. The method relies on operating a set of finite alphabet iterative decoders (FAID). The message-passing update rules are interpreted as discrete dynamical systems, and are judiciously chosen to ensure that decoders have different dynamics on a specific finite-length code. An algorithm is proposed which uses random jumps in the iterative message passing trajectories, such that the system is not trapped in periodic attractors. We show by simulations that the FAID diversity approach with random jumps has the potential of approaching the performance of maximum-likelihood decoding for finite-length regular, column-weight three codes.