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International Conference on Software, Telecommunications and Computer Networks (SoftCOM)

Lowering the error floor of short-to-medium length LDPC codes using optimal low-correlated-edge density (OED) PEG Tanner graphs

Umar-Faruk Abdu-Aguye, Marcel Ambroze, Martin Tomlinson

Published Sep 1, 2016• 1 citation
Publisher

Abstract

We propose a simple modification to the progressive edge-growth (PEG) algorithm in order to control the girth of low-density parity-check (LDPC) code Tanner graphs constructed. Using the modified algorithm, the edge densities in short irregular rate-% LDPC code graphs can be increased without increasing the correlation between edges in graphs by creating cycles of undesired length. Density evolution (DE) optimized symbol node degree distributions used for constructing short-to-medium length codes were further optimized (slightly altered) while maintaining the maximum edge correlation (minimum local girths) that exists in graphs constructed using the unaltered DE optimized distributions. As a result of improved degree sequences (and graph densities) which were derived, short-to-medium length LDPC codes with lower error floors than published for LDPC codes of identical length and rate were found. These slightly denser codes have lower error floor than codes constructed using DE optimized degree distributions in PEG algorithms modified for improved approximate cycle extrinsic message degrees (ACE). Simulation results show that, at a code length of 512, efficient irregular rate-% PEG codes have significantly lower error floors than a rate-% IEEE 802.16e (WiMAX) LDPC code of length 576.

DOI
10.1109/softcom.2016.7772157

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We propose a simple modification to the progressive edge-growth (PEG) algorithm in order to control the girth of low-density parity-check (LDPC) code Tanner graphs constructed. Using the modified algorithm, the edge densities in short irregular rate-% LDPC code graphs can be increased without increasing the correlation between edges in graphs by creating cycles of undesired length. Density evolution (DE) optimized symbol node degree distributions used for constructing short-to-medium length codes were further optimized (slightly altered) while maintaining the maximum edge correlation (minimum local girths) that exists in graphs constructed using the unaltered DE optimized distributions. As a result of improved degree sequences (and graph densities) which were derived, short-to-medium length LDPC codes with lower error floors than published for LDPC codes of identical length and rate were found. These slightly denser codes have lower error floor than codes constructed using DE optimized degree distributions in PEG algorithms modified for improved approximate cycle extrinsic message degrees (ACE). Simulation results show that, at a code length of 512, efficient irregular rate-% PEG codes have significantly lower error floors than a rate-% IEEE 802.16e (WiMAX) LDPC code of length 576.
Gallager proposed and developed low density parity check (LDPC) codes in the early 1960s. LDPC codes were rediscovered in the early 1990s and shown to be capacity approaching over the additive white Gaussian noise (AWGN) channel. Subsequently, density evolution (DE) optimized symbol node degree distributions were used to significantly improve the decoding performance of short to medium length irregular LDPC codes. Currently, the short to medium length LDPC codes with the lowest error floor are DE optimized irregular LDPC codes constructed using progressive edge growth (PEG) algorithm modifications which are designed to increase the approximate cycle extrinsic message degrees (ACE) in the LDPC code graphs constructed. The aim of the present work is to find efficient means to improve on the error floor performance published for short to medium length irregular LDPC codes over AWGN channels in the literature. An efficient algorithm for determining the girth and ACE distributions in short to medium length LDPC code Tanner graphs has been proposed. A cyclic PEG (CPEG) algorithm which uses an edge connections sequence that results in LDPC codes with improved girth and ACE distributions is presented. LDPC codes with DE optimized/’good’ degree distributions which have larger minimum distances and stopping distances than previously published for LDPC codes of similar length and rate have been found. It is shown that increasing the minimum distance of LDPC codes lowers their error floor performance over AWGN channels; however, there are threshold minimum distances values above which there is no further lowering of the error floor performance. A minimum local girth (edge skipping) (MLG (ES)) PEG algorithm is presented; the algorithm controls the minimum local girth (global girth) connected in the Tanner graphs of LDPC codes constructed by forfeiting some edge connections. A technique for constructing optimal low correlated edge density (OED) LDPC codes based on modified DE optimized symbol node degree distributions and the MLG (ES) PEG algorithm modification is presented. OED rate-½ (n, k)=(512, 256) LDPC codes have been shown to have lower error floor over the AWGN channel than previously published for LDPC codes of similar length and rate. Similarly, consequent to an improved symbol node degree distribution, rate ½ (n, k)=(1024, 512) LDPC codes have been shown to have lower error floor over the AWGN channel than previously published for LDPC codes of similar length and rate. An improved BP/SPA (IBP/SPA) decoder, obtained by making two simple modifications to the standard BP/SPA decoder, has been shown to result in an unprecedented generalized improvement in the performance of short to medium length irregular LDPC codes under iterative message passing decoding. The superiority of the Slepian Wolf distributed source coding model over other distributed source coding models based on LDPC codes has been shown.
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.
The progressive-edge-growth (PEG) algorithm for constructing Tanner graphs with large girth by progressively establishing edges/connections between symbol nodes and check nodes in an edge-by-edge way is highly known to construct low-density parity-check (LDPC) codes at short block lengths for achieving good performance. This paper proposed an improved PEG construction for irregular codes, which heuristically selects good codes from random graphs for short block lengths. The simulation results are given to demonstrate that the proposed algorithm gians better BER performance for high signal-to-noise (SNR) ratios without performance loss for low-SNR region, i.e. low error floor.
We propose a general method for constructing Tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge-by-edge manner, called progressive edge-growth (PEG) algorithm. Lower bounds on the girth of PEG Tanner graphs and on the minimum distance of the resulting low-density parity-check (LDPC) codes are derived in terms of parameters of the graphs. Simple variations of the PEG algorithm can also be applied to generate linear-time encodeable LDPC codes. Regular and irregular LDPC codes using PEG Tanner graphs and allowing symbol nodes to take values over GF(q) (q>2) are investigated. Simulation results show that the PEG algorithm is a powerful algorithm to generate good short-block-length LDPC codes.
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.
In this paper, we consider the construction of irregular QC-LDPC codes by jointly optimizing the girth, the number and approximate cycle extrinsic message degree (ACE) of short cycles to further improve the error-floor performance of irregular QC-LDPC codes with given degree distributions. By selecting the best edges, each of which must make the cycles consisting of the current variable node have the best lower bound of ACE or the minimum number of cycles with ACE lower bound, for each variable node via the progressive Edge growth (PEG) approaches, irregular QC-LDPC codes with better tradeoff between girth, the shortest cycle number and ACE could be constructed. Simulation results show that the proposed codes achieve better error-floor performance compared with the codes using the same degree distributions constructed by the recent proposed methods.