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IEEE Transactions on Communications

Constructing Short-Length Irregular LDPC Codes with Low Error Floor

Xia Zheng, Francis C. M. Lau, Chi Kong Tse

Published Sep 20, 2010• 61 citations
Publisher

Abstract

Trapping sets (TSs) are known to cause error floors in regular and irregular low-density parity-check (LDPC) codes. By avoiding major error-contributing TSs during the code construction process, codes with low error floors can effectively be built. In it has been shown that TSs labeled as [w;u] are considered as being equivalent under the automorphism of the graph and are therefore contributing equally to the error floor. However, TSs with the same label [w;u] are not identical in general, particularly for the case of irregular LDPC codes. In this paper, we introduce a parameter e that can identify the number of "distinguishable" cycles in the connected subgraph induced by an elementary trapping set. Further, we propose a code construction algorithm, namely the Progressive-Edge-Growth Approximate-minimum-Cycle-Set-Extrinsic-message-degree (PEG-ACSE) method, that aims to avoid small elementary trapping sets (ETSs), particularly detrimental ETSs. We also develop theorems evaluating the minimum possible ETSs formed by PEG construction algorithms in general. We compare the characteristics of the codes built using the proposed method and those built using PEG-only or PEG-Approximate-minimum-Cycle-Extrinsic-message-degree (PEG-ACE) methods. Results from simulations show that the codes constructed using the proposed PEG-ACSE method produce lower error rates, particularly at the high signal-to-noise (SNR) region, compared with codes constructed using other PEG-based algorithms.

DOI
10.1109/tcomm.2010.083110.080638

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Trapping sets (TSs) are known to cause error floors in regular and irregular low-density parity-check (LDPC) codes. By avoiding major error-contributing TSs during the code construction process, codes with low error floors can effectively be built. In it has been shown that TSs labeled as [w;u] are considered as being equivalent under the automorphism of the graph and are therefore contributing equally to the error floor. However, TSs with the same label [w;u] are not identical in general, particularly for the case of irregular LDPC codes. In this paper, we introduce a parameter e that can identify the number of "distinguishable" cycles in the connected subgraph induced by an elementary trapping set. Further, we propose a code construction algorithm, namely the Progressive-Edge-Growth Approximate-minimum-Cycle-Set-Extrinsic-message-degree (PEG-ACSE) method, that aims to avoid small elementary trapping sets (ETSs), particularly detrimental ETSs. We also develop theorems evaluating the minimum possible ETSs formed by PEG construction algorithms in general. We compare the characteristics of the codes built using the proposed method and those built using PEG-only or PEG-Approximate-minimum-Cycle-Extrinsic-message-degree (PEG-ACE) methods. Results from simulations show that the codes constructed using the proposed PEG-ACSE method produce lower error rates, particularly at the high signal-to-noise (SNR) region, compared with codes constructed using other PEG-based algorithms.
The progressive-edge-growth (PEG) algorithm of Hu et al. is modified to improve the error floor performance of the constructed low-density parity-check (LDPC) codes. To improve the error floor, the original PEG algorithm is equipped with an efficient algorithm to find the dominant elementary trapping sets (ETS's) that are added to the Tanner graph of the under-construction code by the addition of each variable node and its adjacent edges. The aim is to select the edges, among the candidates available at each step of the original PEG algorithm, that prevent the creation of dominant ETS's. The proposed method is applicable to both regular and irregular variable node degree distributions. Simulation results are presented to demonstrate the superior ETS distribution and error floor performance of the constructed codes compared to similar codes constructed by the original and other modifications of the PEG algorithm.
Trapping sets (TSs) have been known to contribute to errors in the decoding of low-density parity-check (LDPC) codes, particularly at the high signal-to-noise ratio (SNR) region. Moreover, TSs with the same label [w; u] are considered equivalent under the automorphism of the graph of a regular code. But according to our our simulations, TSs with the same label [w; u] are producing different error rates in the case of irregular codes. In this paper, we will identify and explain the cause of the differences in error rates for TSs with the same label. Further, we will propose a simple mechanism to differentiate these TSs.
It has been known that stopping sets and trapping sets are the main contributors to error floors exhibited by short-length low-density parity-check (LDPC) codes. In this work, a new metric called ldquoapproximate cycle set extrinsic message degree (ACSE)rdquo is defined to help removing stopping sets with small size and bad trapping sets. Based on the new metric, we propose a code-construction algorithm that produces LDPC codes with very low error floors. Finally, we compare the error rates between codes produced by the proposed algorithm and MacKay code.
In this paper, we study the graphical structure of elementary trapping sets (ETS) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree dland girth g, we identify all the non-isomorphic structures of an arbitrary class of (a, b) ETSs, where a is the number of variable nodes and b is the number of odd-degree check nodes in the induced subgraph of the ETS. Our study leads to a simple characterization of dominant classes of ETSs (those with relatively small values of a and b) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set S in the class is a layered superset (LSS) of a short cycle, where the term “layered” is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate S. This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for dl= 3, 4, 5, 6, g = 6, 8 and a, b ≤ 10. The results of this work can be used for the error floor analysis and for the design of LDPC codes with low error floors.
In this paper, we study the graphical structure of elementary trapping sets (ETSs) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree dl and girth g, we identify all the nonisomorphic structures of an arbitrary class of (a, b) ETSs, where a is the number of variable nodes and b is the number of odd-degree check nodes in the induced subgraph of the ETS. This paper leads to a simple characterization of dominant classes of ETSs (those with relatively small values of a and b) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set S in the class is a layered superset (LSS) of a short cycle, where the term layered is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate S. This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for dl= 3, 4, 5, 6, g = 6, 8, and a, b ≤ 10 in this paper. The results of this paper can be used for the error floor analysis and for the design of LDPC codes with low error floors.
In this paper we introduce a novel progressive edge-growth (PEG) algorithm, the cyclic PEG (CPEG) algorithm. The CPEG algorithm uses an alternative edge establishment sequence to construct low-density parity-check (LDPC) codes. Irregular LDPC codes constructed using the CPEG algorithm have improved girth and approximate cycle extrinsic message degree (ACE) compared to existing PEG algorithms. We also analyze the minimum codeword weight, minimum stopping set weight, local girth, and local ACE distributions for codes in four very large ensembles of irregular LDPC codes. The code ensembles analyzed were constructed using standard PEG, ACE modified standard PEG, CPEG, and ACE modified CPEG algorithms. Modifications to improve the ACE in PEG LDPC codes, by Xiao and Banihashemi, were implemented in the ‘ACE modified’ versions of the PEG algorithms. The ACE modified standard PEG algorithm constructed the code ensemble with the highest minimum codeword weight and minimum stopping set weight distributions, and the ACE modified CPEG algorithm constructed the code ensemble with the highest local girth and ACE distributions. Short block length irregular LDPC codes with good degree distributions which have higher minimum weights than have been published for similar LDPC codes were found in the four code ensembles.