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International Conference on Communications

Constructing LDPC codes from simple loop-free encoding modules

D. Divsalar, Samuel J. Dolinar, Jeremy Thorpe, C. Jones

Published Aug 16, 2005• 30 citations
Publisher

Abstract

Inspired by recently proposed accumulate-repeat-accumulate (ARA) codes, in this paper we propose a construction method for LDPC codes using simple loop-free encoding modules. Such codes can be viewed as serial/parallel concatenations of simple modules such as accumulators, repetition codes, differentiators, and punctured single parity check codes. Examples are accumulate-repeat-accumulate (ARA) codes, accumulate-repeat-accumulate-accumulate (ARAA) codes and accumulate-repeat-check-accumulate codes, and other variations. These codes constitute a subclass of LDPC codes with very fast encoder structure. They also have a projected graph or protograph representation that allows for high-speed decoder implementation. Based on density evolution, we show through some examples that low iterative decoding thresholds close to the channel capacity limits can be achieved with low maximum variable node degrees, as the block size goes to infinity. The decoding threshold in many examples outperforms that of the best known unstructured irregular LDPC codes constrained to have the same maximum node degree. Furthermore, by puncturing the accumulator modules, any desired higher rate codes can be obtained with thresholds that stay close to their respective channel capacity thresholds uniformly.

DOI
10.1109/icc.2005.1494433

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Inspired by recently proposed accumulate-repeat-accumulate (ARA) codes, in this paper we propose a construction method for LDPC codes using simple loop-free encoding modules. Such codes can be viewed as serial/parallel concatenations of simple modules such as accumulators, repetition codes, differentiators, and punctured single parity check codes. Examples are accumulate-repeat-accumulate (ARA) codes, accumulate-repeat-accumulate-accumulate (ARAA) codes and accumulate-repeat-check-accumulate codes, and other variations. These codes constitute a subclass of LDPC codes with very fast encoder structure. They also have a projected graph or protograph representation that allows for high-speed decoder implementation. Based on density evolution, we show through some examples that low iterative decoding thresholds close to the channel capacity limits can be achieved with low maximum variable node degrees, as the block size goes to infinity. The decoding threshold in many examples outperforms that of the best known unstructured irregular LDPC codes constrained to have the same maximum node degree. Furthermore, by puncturing the accumulator modules, any desired higher rate codes can be obtained with thresholds that stay close to their respective channel capacity thresholds uniformly.
This paper provides a construction method for low-rate low density parity check codes. Inspired by recently proposed accumulate-repeat-accumulate (ARA) codes, and hybrid concatenated codes, in this paper we extend the construction to low rates. Such codes can be viewed as hybrid concatenations of simple modules such as accumulators, repetition codes, differentiators, and punctured single parity check codes. These codes constitute a subclass of LDPC codes with very fast encoder structure. They also have a projected graph or protograph representation that allows for high-speed decoder implementation. Based on density evolution, we show through some examples that low iterative decoding thresholds close to the channel capacity limits can be achieved, as the block size goes to infinity. Iterative decoding simulation results for short blocks are provided for a few examples that show near-capacity performance and very low error floor
Accumulate-repeat-accumulate-accumulate (ARAA) codes have been proposed, inspired by the recently proposed accumulate-repeat-accumulate (ARA) codes. These are error-correcting codes suitable for use in a variety of wireless data-communication systems that include noisy channels. ARAA codes can be regarded as serial turbolike codes or as a subclass of low-density parity-check (LDPC) codes, and, like ARA codes they have projected graph or protograph representations; these characteristics make it possible to design high-speed iterative decoders that utilize belief-propagation algorithms. The objective in proposing ARAA codes as a subclass of ARA codes was to enhance the error-floor performance of ARA codes while maintaining simple encoding structures and low maximum variable node degree.
We propose several LDPC code constructions that simultaneously achieve good threshold and error floor performance. Minimum distance is shown to grow linearly with block size (similar to regular codes of variable degree at least 3) by considering ensemble average weight enumerators. Our constructions are based on projected graph, or protograph, structures that support high-speed decoder implementations. As with irregular ensembles, our constructions are sensitive to the proportion of degree-2 variable nodes. A code with too few such nodes tends to have an iterative decoding threshold that is far from the capacity threshold. A code with too many such nodes tends to not exhibit a minimum distance that grows linearly in block length. In this paper we also show that preceding can be used to lower the threshold of regular LDPC codes. The decoding thresholds of the proposed codes, which have linearly increasing minimum distance in block size, outperform that of regular LDPC codes. Furthermore, a family of low to high rate codes, with thresholds that adhere closely to their respective channel capacity thresholds, is presented. Simulation results for a few example codes show that the proposed codes have low error floors as well as good threshold SNR performance.
This work investigated the performance of a newly developed protograph-based low-density parity-check code (LDPC) families belonging to accumulate-repeat-4-jagged-accumulate codes. The proposed code has hierarchical irregularity and quasi-cyclic property that retains it to be suitable for fast encoder and decoder implementation. The code used in this work is a rate-compatible linearly punctured code. The modified iterative decoding algorithm as mentioned in the literature is used along with a newly proposed encoder and simulation has been done for proposed LDPC coded orthogonal frequency division multiplexing (OFDM) link over wireless and optical wireless channels with hard- and soft-decision decoding. The code relates to new methods for designing rate-compatible families of protographs of different rates, all having the linear minimum distance property showing small performance degradation compared with the threshold. The parity-check matrix is composed of small identical modules to reduce the routing complexity of the decoder. The main essence of this code lies in the fact that LDPC codes of arbitrarily large size can be built by expanding from individual protographs or rate-compatible protograph families.
Recently, a new class of Low-Density Parity-Check (LDPC) codes constructed from a template was introduced. This kind of codes were called protograph codes. The protograph serves as a blueprint for constructing LDPC codes of arbitrary size whose performance can be predicted by analyzing the protograph. Protograph LDPC codes perform very well and suit for high-speed encoding and decoding. However, there are few researches on expansion and encoding algorithms for them. In this paper, using isomorphism between matrix ring and polynomial ring, we propose efficient quasi-cyclic expansion and encoding algorithms for protograph LDPC codes. Simulation results show that the protograph LDPC codes which were constructed by the algorithm proposed in this paper, outperform the best known unstructured irregular LDPC codes with the same maximum node degrees.
A method is presented for constructing LDPC codes with excellent performance, simple hardware implementation, low encoder complexity, and which can be concisely documented. The simple code structure is achieved by using a base graph, expanded with circulants. The base graph is chosen by computer search using simulated annealing, driven by density evolution's decoding threshold as determined by the reciprocal channel approximation. To build a full parity check matrix, each edge of the base graph is replaced by a circulant permutation, chosen to maximize loop length by using a Viterbi-like algorithm.