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IEEE International Symposium on Information Theory (ISIT)

Design of an LDPC code with low error floor

Sang Hyun Lee, Kwang Soon Kim, Jae Kyun Kwon, Yun Hee Kim, Jae Young Ahn

Published Jan 1, 2005• 5 citations
Publisher

Abstract

A search algorithm for stopping sets in a Tanner graph is proposed for designing good low-density parity-check (LDPC) codes. By applying the belief-propagation algorithm with messages containing the information of originated variable nodes and their connected edges, the stopping sets can be detected. Furthermore, a code design method using the algorithm is presented and the performances of the designed code over several channels are shown

DOI
10.1109/isit.2005.1523486

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IEEE Transactions on Information Theory•Jul 24, 2008•115 citations
A search algorithm for stopping sets in a Tanner graph is proposed for designing good low-density parity-check (LDPC) codes. By applying the belief-propagation algorithm with messages containing the information of originated variable nodes and their connected edges, the stopping sets can be detected. Furthermore, a code design method using the algorithm is presented and the performances of the designed code over several channels are shown
A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding threshold is as large as possible. We introduce a new code optimization technique which involves the search space range which can be thought of as minimizing randomness in differential evolution or limiting the search range in exhaustive search. This technique is applied to the design of good irregular LDPC codes and multi-edge type LDPC codes.
We present a method for lowering the error floor of low-density parity check (LDPC) codes. It is based on Tanner graph covers that do not have trapping sets from the original code. The advantages of the method are that it is universal, as it can be applied to any LDPC code/channel model/decoding algorithm and it improves performance at the expense of increasing the code length, without losing the code regularity, without changing the decoding algorithm, and, under certain conditions, without lowering the code rate. We illustrate the method by modifying Tanner, MacKay and Margulis codes to improve performance on the binary symmetric channel (BSC) under the Gallager B decoding algorithm.
There are many algorithms to construct good low-density parity-check (LDPC) code, the most typical algorithms are bit-filling algorithm, randomly construction algorithm, and PEG (progressive edge growth) algorithm. As shown in [1], the error floor of LDPC Code is decreased by using PEG algorithm, but the error correction performance in waterfall region is compromised since a stopping set with small size will form the codeword with small Hamming weight over AWGN [2]. In this paper, we propose a novel algorithm to construct LDPC Codes. In our algorithm, we construct LDPC Code to avoid small girth and small stopping set by detecting the complete associated matrix of check node (defined in this paper) that converted from bipartite graph of LDPC Code based on the graph theory. Simulation shows that the LDPC code constructed by our algorithm has lower error floor than randomly constructed LDPC Code. The performance improvement of our algorithm is 0.1dB at BER of 10-3compared with PEG algorithm.
Low-density parity-check (LDPC) codes can approach the Shannon limit performance closely, and are becoming one of the most promising channel codes in the Error Control Coding area.The performance of an LDPC code with given length is mainly affected by the sizes of some combinatorial characteristics of its corresponding bipartite graph and the distribution of variable degrees and check node degrees.With the help of the density evolution algorithm, randomly designed LDPC codes, with carefully chosen degree distribution pairs, have been shown to achieve better Shannon capacity performance than their regular counterparts when decoded using the iterative belief propagation (BP) decoding algorithm.However, regular LDPC codes outperform their irregular counterparts in term of their error floors.Therefore, construction of LDPC codes which have both attractive error rate performance and low error floor performance has become an attractive topic.Currently, optimization of variable node degree distribution and check node degree distribution for an LDPC code with given length, which could help the code to achieve good Shannon limit performance, has been extensively studied.In this dissertation, we demonstrate some approaches of improving the error floor performance for a given LDPC code with desired degree distribution pair.The contribution mainly includes three parts.In the first part, the relationship between cycles and unfavorable combinatorial
The performance of low-density parity-check (LDPC) codes over the binary erasure channel for a low erasure probability is limited by small stopping sets. The frame error rate over the additive white Gaussian noise (AWGN) channel for a high signal-to-noise ratio is limited by small trapping sets and by the minimum distance of the LDPC code, which is equal to the codeword with the minimal Hamming weight. In this paper, we present a simple and fast algorithm based on the error impulse method to find small stopping sets in the Tanner graphs of LDPC codes. Since a codeword with small Hamming weight can be represented by a special stopping set, the algorithm can also be used to find the minimum distance of LDPC codes. Furthermore, we use the size of the small stopping sets and the minimum distance of the code to calculate asymptotic performance bounds for LDPC codes transmitted over the binary erasure and the AWGN channel, respectively and compare the bounds with simulations. 1
We discuss error floor asympotics and present a method for improving the performance of low-density parity-check (LDPC) codes in the high SNR (error floor) region. The method is based on Tanner graph covers that do not have trapping sets from the original code. The advantages of the method are that it is universal, as it can be applied to any LDPC code/channel/decoding algorithm and it improves performance at the expense of increasing the code length, without losing the code regularity, without changing the decoding algorithm, and, under certain conditions, without lowering the code rate. The proposed method can be modified to construct convolutional LDPC codes also. The method is illustrated by modifying Tanner, MacKay and Margulis codes to improve performance on the binary symmetric channel (BSC) under the Gallager B decoding algorithm. Decoding results on AWGN channel are also presented to illustrate that optimizing codes for one channel/decoding algorithm can lead to performance improvement on other channels.