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On q-ary LDPC Code Design for a Low Error Floor

Andrea Marinoni, Pietro Savazzi, Richard D. Wesel

Published Dec 1, 2011• 1 citation
Publisher

Abstract

This paper explores protograph-based and ACE-based methods for constructing q-ary low-density parity-check (LDPC) matrices. The ACE approach maximizes approximate cycle extrinsic message degree, explicitly avoiding small q-ary stopping sets and implicitly avoiding small absorbing sets. In addition to ACE, this paper applies linear-dependent-set maximization (LDSM) to the binary image of the q-ary LDPC matrix. Performance is studied for binary and q-ary instances of erasure channels and additive white Gaussian noise channels. The combination of the ACE approach and LDSM provides dramatic error floor improvement for the binary erasure channel and both binary and q-ary AWGN channels.

DOI
10.1109/glocom.2011.6134057

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This paper explores protograph-based and ACE-based methods for constructing q-ary low-density parity-check (LDPC) matrices. The ACE approach maximizes approximate cycle extrinsic message degree, explicitly avoiding small q-ary stopping sets and implicitly avoiding small absorbing sets. In addition to ACE, this paper applies linear-dependent-set maximization (LDSM) to the binary image of the q-ary LDPC matrix. Performance is studied for binary and q-ary instances of erasure channels and additive white Gaussian noise channels. The combination of the ACE approach and LDSM provides dramatic error floor improvement for the binary erasure channel and both binary and q-ary AWGN channels.
We propose a technique to design finite-length irregular low-density parity-check (LDPC) codes over the binary-input additive white Gaussian noise (AWGN) channel with good performance in both the waterfall and the error floor region. The design process starts from a protograph which embodies a desirable degree distribution. This protograph is then lifted cyclically to a certain block length of interest. The lift is designed carefully to satisfy a certain approximate cycle extrinsic message degree (ACE) spectrum. The target ACE spectrum is one with extremal properties, implying a good error floor performance for the designed code. The proposed construction results in quasi-cyclic codes which are attractive in practice due to simple encoder and decoder implementation. Simulation results are provided to demonstrate the effectiveness of the proposed construction in comparison with similar existing constructions.
We propose a technique to design finite-length irregular low-density parity-check (LDPC) codes over the binary-input additive white Gaussian noise (AWGN) channel with good performance in both the waterfall and the error floor region. The design process starts from a protograph which embodies a desirable degree distribution. This protograph is then lifted cyclically to a certain block length of interest. The lift is designed carefully to maximize the components of the approximate cycle extrinsic message degree (ACE) spectrum of the code's Tanner graph in a greedy fashion. As a consequence, the designed code would perform well in the error floor region. Moreover, the proposed construction results in quasi-cyclic codes which are attractive in practice due to simple encoder and decoder implementation. Simulation results are provided to demonstrate the effectiveness of the proposed construction in comparison with similar existing constructions.
To obtain efficient and pragmatic channel codes high signal-to-noise ratios (SNRs), an optimized construction of short and high rate protograph quasi-cyclic (QC) low-density parity-check (LDPC) codes is presented. Based on an optimized protograph template, the code framework is firstly produced by the extensions of the variable nodes. By enlarging the dimension of the sub-matrices related to the protograph framework, the base quasi-cyclic (QC) matrix is generated with required code rate and length. Then, the elements in the base matrix are split with even smaller square sub-matrices of the same row and column weights. In this procedure, a progressive edge growth (PEG) algorithm is employed to find the optimized positions of the QC submatrices, which brings larger girth for better code performance. Also the circulant offsets in all QC sub-matrices are optimally searched by a QC improved approximated cycle extrinsic message degree (QCI-ACE) algorithm, which improves the relationship of the unavoidable loops in the code's Tanner graph and thus it improves the error floor. Simulation results show that the codes produced by the proposed method demonstrate quite good bit error rate (BER) performance. In addition, they exhibit good properties of high spectrum efficiency by the high code rate, as well as the low complexity by the short code length. Therefore, the proposed code construction can be efficiently applied in the design of short and high rate protograph QC-LDPC codes with high rate and rational complexity.
Long quasi-cyclic low-density parity check (QC-LDPC) codes have been chosen as one of the forward error correction (FEC) schemes in passive optical network (PON) systems benefiting from the excellent performance. About how to get the appropriate LDPC codes, some traditional methods which focus on obtaining large girth or flat approximated cycle extrinsic message degree (ACE) spectrums are proposed for the design of LDPC codes. These methods which may still make the existence of small stopping sets containing short cycles probably are not so useful for long QC-LDPC codes. Recently, the parallel vector message passing (PMP) algorithm oriented-to decreasing the number of short cycles is proved to be an effective method to optimize LDPC codes. Since ignoring short cycles with poor connectivity which can induce error floor, this method cannot guarantee its effectiveness for long QC-LDPC codes. In this letter, an edge exchange optimization method based on the global properties of ACE and girth called EEA method is proposed, which tries to reduce cycles with low ACE in the order of cycle length. The numerical results prove that our EEA method can bring the improvements of error floor up to two orders of magnitude to long QC-LDPC codes and obviously outperforms PMP algorithm.
This paper presents an efficient algorithm for finding the dominant trapping sets of a low-density parity-check (LDPC) code. The algorithm can be used to estimate the error floor of LDPC codes or as a tool to design LDPC codes with low error floors. For regular codes, the algorithm is initiated with a set of short cycles as the input. For irregular codes, in addition to short cycles, variable nodes with low degree and cycles with low approximate cycle extrinsic message degree (ACE) are also used as the initial inputs. The initial inputs are then expanded recursively to dominant trapping sets of increasing size. At the core of the algorithm lies the analysis of the graphical structure of dominant trapping sets and the relationship of such structures to short cycles, low-degree variable nodes, and cycles with low ACE. The algorithm is universal in the sense that it can be used for an arbitrary graph and that it can be tailored to find a variety of graphical objects, such as absorbing sets and Zyablov-Pinsker trapping sets, known to dominate the performance of LDPC codes in the error floor region over different channels and for different iterative decoding algorithms. Simulation results on several LDPC codes demonstrate the accuracy and efficiency of the proposed algorithm. In particular, the algorithm is significantly faster than the existing search algorithms for dominant trapping sets.
This work explains the relationship between cycles, stopping sets, and dependent columns of the parity check matrix of low-density parity-check (LDPC) codes. Furthermore, it discusses how these structures limit LDPC code performance under belief propagation decoding. A new metric called extrinsic message degree (EMD) measures cycle connectivity in bipartite graph. Using an easily computed estimate of EMD, we propose a Viterbi-like algorithm that selectively avoids cycles and increases stopping set size. This algorithm yields codes with error floors that are orders of magnitude below those of girth-conditional codes.