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IEEE Transactions on Communications

Tail-Biting Globally-Coupled LDPC Codes

Ji Zhang, Baoming Bai, Shuangyang Li, Min Zhu, Huaan Li

Published Oct 1, 2019• 18 citations
Publisher

Abstract

This paper presents a new type of globally-coupled low-density parity-check (GC-LDPC) codes whose base matrix has a cyclic structure in the global part. Therefore, the resulting codes are referred to as tail-biting GC-LDPC (TB-GC-LDPC) codes. We propose two methods to construct TB GC quasi-cyclic LDPC (TB-GC-QC-LDPC) codes. For the first method, we extract a replicated version of a constructed base matrix and mask it with a designed masking matrix. Compared to the conventional construction methods, this method provides more flexibility in code length for TB-GC-QC-LDPC codes. The second method is based on designing the incidence matrix of a special type of packings. Examples show that the constructed TB-GC-QC-LDPC codes perform well over the additive white Gaussian noise channel (AWGNC) and the binary erasure channel (BEC). The asymptotic performance of TB-GC-LDPC ensembles over BECs are also analyzed by resorting to density evolution. Moreover, numerical results show that TB-GC-LDPC ensembles can achieve better flooding-schedule decoding (FSD) thresholds than the corresponding GC-LDPC ensembles with a similar structure. With sufficient decoding iterations in the global phase, the local/global two-phase iterative decoding (TPD) thresholds of the TB-GC-LDPC ensembles significantly outperform those of the corresponding GC-LDPC ensembles as well.

DOI
10.1109/tcomm.2019.2944621

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This paper presents a new type of globally-coupled low-density parity-check (GC-LDPC) codes whose base matrix has a cyclic structure in the global part. Therefore, the resulting codes are referred to as tail-biting GC-LDPC (TB-GC-LDPC) codes. We propose two methods to construct TB GC quasi-cyclic LDPC (TB-GC-QC-LDPC) codes. For the first method, we extract a replicated version of a constructed base matrix and mask it with a designed masking matrix. Compared to the conventional construction methods, this method provides more flexibility in code length for TB-GC-QC-LDPC codes. The second method is based on designing the incidence matrix of a special type of packings. Examples show that the constructed TB-GC-QC-LDPC codes perform well over the additive white Gaussian noise channel (AWGNC) and the binary erasure channel (BEC). The asymptotic performance of TB-GC-LDPC ensembles over BECs are also analyzed by resorting to density evolution. Moreover, numerical results show that TB-GC-LDPC ensembles can achieve better flooding-schedule decoding (FSD) thresholds than the corresponding GC-LDPC ensembles with a similar structure. With sufficient decoding iterations in the global phase, the local/global two-phase iterative decoding (TPD) thresholds of the TB-GC-LDPC ensembles significantly outperform those of the corresponding GC-LDPC ensembles as well.
The masking matrix plays an important role in constructing new classes of regular and irregular quasi-cyclic low density parity check (QC-LDPC) codes. By coupling two identical graphs in a special way, we present a new structure of the masking matrix, whose Tanner graph can be seen as a protograph. From this perspective, we propose a Gaussian Approximation algorithm for protograph-based LDPC codes to analyze the asymptotic performance of this class of codes. It is shown that, although the proposed structure of the masking matrix has almost the same convergence threshold as the conventional one produced randomly by progressive edge growth (PEG) algorithm, the former converges faster than the latter. Simulation results illustrate that the QC-LDPC code constructed by the proposed structure of the masking matrix has about 0.2dB coding gains compared with the conventional one.
A novel method of low-complexity near-maximum-likelihood (ML) decoding of quasi-cyclic (QC) low-density parity-check (LDPC) codes over the binary erasure channel is presented. The idea is similar to wrap-around decoding of tail-biting convolutional codes. ML decoding is applied to a relatively short window which is cyclically shifted along the received sequence. The procedure is repeated until either all erasures have been corrected, or no new erasures are corrected at a certain round. A new upper bound on the ensemble-average ML decoding error probability for a finite-length row-regular LDPC code family is derived and presented. Furthermore, a few examples of regular and irregular QC LDPC codes are studied by simulations and their performance is compared with the ensemble-average performance. Finally, the impact of the codeword weight and stopping set size spectra on the ML and belief-propagation decoding performance is discussed.
This paper presents a special type of LDPC codes with a structure related to but different from that of the spatially coupled LDPC codes. For an LDPC code of this type, its Tanner graph is composed of a set of small disjoint Tanner graphs which are connected together by a group of overall check-nodes, called global check-nodes. Codes of this type are called globally coupled LDPC codes and they perform well over both the additive Gaussian white noise and the binary-erasure channels. Furthermore, they are very effective at correcting erasures clustered in bursts. Two algebraic methods are presented for constructing these codes. A two-phase local/global iterative scheme for decoding these codes is presented. This decoding scheme allows correction of local random and global errors and/or erasures in two phases.
Low-Density Parity-Check (LDPC) codes are among the most effective modern error-correcting codes due to their excellent correction performance and highly parallel decoding scheme. Moreover, the nonbinary extension of such codes further increases performance in the short-block regime. In this paper, we review the key elements for the construction of implementation-oriented binary and nonbinary codes. These Quasi-Cyclic LDPC (QC-LDPC) codes additionally feature efficient encoder and decoder implementation frameworks. We then present a versatile algorithm for the construction of both binary and nonbinary QC-LDPC codes that have low encoding complexity and an optimized corresponding graph structure. Our algorithm uses a progressive edge growth algorithm, modified for QC-LDPC graph construction, and then performs an iterative global search for optimized cyclic shift values within the QC-LDPC circulants. Strong error correction performance is achieved by minimizing the number of short cycles, and cycles with low external connectivity, within the code graph. We validate this approach via error rate simulations of a transmission system model featuring an LDPC coder-decoder, digital modulation, and additive white Gaussian noise channels. The obtained numerical results validate the effectiveness of the proposed construction algorithm, with a number of constructed codes exhibiting either similar or superior performance to industry standard binary codes and selected nonbinary codes from the literature.
This paper presents a special type of LDPC codes constructed based on the conventional parity-check matrices of Reed-Solomon (RS) codes. LDPC codes of this type are referred to as globally coupled RS-LDPC codes. These codes are designed for correcting random errors, random short phased bursts of erasures and long bursts of erasures. The Tanner graph of a globally coupled RS-LDPC is composed of a number of disjoint copies of the Tanner graph of a local RS-LDPC code which are connected together by a group of overall check-nodes (CNs), called global CNs. A globally coupled RS-LDPC code can be decoded with a two-phase local/global iterative scheme which allows correction of local random errors and phased bursts of erasures in the local phase and global random errors and a single long burst of erasures in the global phase.