Scientists have competed to find codes that can be decoded with optimal decoding algorithms. Generalized LDPC codes were found to compare well with such codes. LDPC codes are well treated with both types of decoding; HDD and SDD. On the other hand GLDPC codes iterative decoding, on both AWGN and BSC channels, was not sufficiently investigated in the literature. This chapter first describes its construction then discusses its iterative decoding algorithms on both channels so far. The SISO decoders, of GLDPC component codes, show excellent error performance with moderate and high code rate. However, the complexities of such decoding algorithms are very high. When the HDD BF algorithm presented to LDPC for its simplicity and speed, it was far from the BSC capacity. Therefore involving LDPC codes in optical systems using such algorithms is a wrong choice. GLDPC codes can be introduced as a good alternative of LDPC codes as their performance under BF algorithm can be improved and they would then be a competitive choice for optical communications. This chapter will discuss the iterative HDD algorithms that improve decoding error performance of GLDPC codes. SDD algorithms that maintain the performance but lowering decoding simplicity are also described.
Scientists have competed to find codes that can be decoded with optimal decoding algorithms. Generalized LDPC codes were found to compare well with such codes. LDPC codes are well treated with both types of decoding; HDD and SDD. On the other hand GLDPC codes iterative decoding, on both AWGN and BSC channels, was not sufficiently investigated in the literature. This chapter first describes its construction then discusses its iterative decoding algorithms on both channels so far. The SISO decoders, of GLDPC component codes, show excellent error performance with moderate and high code rate. However, the complexities of such decoding algorithms are very high. When the HDD BF algorithm presented to LDPC for its simplicity and speed, it was far from the BSC capacity. Therefore involving LDPC codes in optical systems using such algorithms is a wrong choice. GLDPC codes can be introduced as a good alternative of LDPC codes as their performance under BF algorithm can be improved and they would then be a competitive choice for optical communications. This chapter will discuss the iterative HDD algorithms that improve decoding error performance of GLDPC codes. SDD algorithms that maintain the performance but lowering decoding simplicity are also described.
In modern communication systems Low-Density Parity Check (LDPC) codes are widely used. It is because of their performance under iterative decoding, hardware-friendly structure and well-known construction methods. For one to construct good LDPC code Density Evolution (DE) can be used, it estimates the decoding threshold, that is it estimates performance of the code. LDPC codes can be further extended to Generalized LDPC (GLDPC) codes, where simple SPC code is replaced with some component code. With more complex structure common DE is no longer correct. This letter presents the upper bound for decoding error and, therefore, an upper bound of a iterative threshold (in dB). It will be shown that the Gaussian approximation gives an upper bound for a decoding error. As one of the results it means that EXIT-charts analysis also gives an upper bound for an iterative threshold for a GLDPC code.
Low-density parity-check (LDPC) codes have received significant attention due to their near-Shannon-limit error performance. The belief propagation (BP) algorithm is the best-known decoding algorithm for LDPC codes. However it often needs several tens or hundreds of iterations to converge. Although fast convergence BP algorithms have been proposed, there is neither an easy tool to analyze their performance nor a standard VLSI architecture to implement them. Moreover, there exists a trade-off between waterfall (i.e., threshold) and error floor (i.e., minimum pseudo distance) performances for LDPC codes. For generalized LDPC (GLDPC) codes, better minimum distance properties can be achieved but at the expense of a rate loss. This research investigates performance analysis and hardware implementation of LDPC codes using fast convergence algorithms. Closed-form extrinsic mutual information transfer (EXIT) functions for several fast convergence BP decoding algorithms are proposed. It is shown theoretically that these algorithms converge faster than standard BP, while the corresponding thresholds remain unchanged. Since these algorithms are of serial nature, their parallelization without compromising convergence speed and error performance is also investigated. This result is useful in hardware implementation because in practice partly parallel decoding schemes are often used. This result also sheds some light on how to group nodes in order to preserve performance. The VLSI architecture of a particular decoding algorithm for LDPC codes is proposed. This approach can be adopted in many standards to achieve a large throughput. Doubly GLDPC (DGLDPC) codes are proposed to compensate the rate loss of GLDPC codes. EXIT charts are used to analyze DGLDPC codes and EXIT functions for variable component codes are thoroughly discussed. Based on EXIT charts, differential evolution is used to optimize their threshold. Ensemble weight enumerators are generalized to protograph-based DGLDPC codes. Simulation results show that DGLDPC codes constructed based on these results can improve the performance of their LDPC and GLDPC counterparts in both the waterfall region and the error floor region.
A generalized low-density parity check code (GLDPC) is a low-density parity check code in which the constraint nodes of the code graph are block codes, rather than single parity checks. In this paper, we study GLDPC codes which have BCH or Reed-Solomon codes as subcodes under bounded distance decoding (BDD). The performance of the proposed scheme is investigated in the limit case of an infinite length (cycle free) code used over a binary erasure channel (BEC) and the corresponding thresholds for iterative decoding are derived. The performance of the proposed scheme for finite code lengths over a BEC is investigated as well. Structures responsible for decoding failures are defined and a theoretical analysis over the ensemble of GLDPC codes which yields exact bit and block error rates of the ensemble average is derived. Unfortunately this study shows that GLDPC codes do not compare favorably with their LDPC counterpart over the BEC. Fortunately, it is also shown that under certain conditions, objects identified in the analysis of GLDPC codes over a BEC and the corresponding theoretical results remain useful to derive tight lower bounds on the performance of GLDPC codes over a binary symmetric channel (BSC). Simulation results show that the proposed method yields competitive performance with a good decoding complexity trade-off for the BSC.
Generalized low-density parity-check (GLDPC) codes have emerged as a promising class of error-correcting codes that extend the capabilities of traditional LDPC codes by incorporating more powerful component codes. This paper investigates an approach to construct long GLDPC codes using short component codes. In this scheme, the BCJR algorithm is used as a component decoder and integrated into the belief propagation algorithm. Simulation results demonstrate that constructions incorporating quasi-cyclic LDPC codes yield the best performance. Compared to the conventional decoding approach for LDPC codes, the proposed generalized constructions demonstrated better bit and frame error rate performance. This approach is promising for communication systems that require low latency. A comparison is also provided with LDPC codes constructed using the Progressive Edge-Growth (PEG) algorithm.
Finding coding schemes that approach the Shannon capacity limit has been the central theme during the development of error-control coding (ECC).In recent years low-density parity-check (LDPC) codes, due to their remarkable performance under iterative decoding, have received extensive study and have been adopted by many modern advanced communication systems.LDPC codes can be further extended to generalized LDPC (GLDPC) codes and doublygeneralized LDPC (DGLDPC) codes.GLDPC and DGLDPC codes are supersets of LDPC codes that possess more generic code structures.GLDPC and DGLDPC codes are more flexible than LDPC codes, since a richer variety of check nodes (CNs) or variable nodes (VNs) offers additional options and flexibility, which enable many advantages over LDPC codes in numerous scenarios.In this dissertation we focus on improved design methods for finite-length (FL) GLDPC and DGLDPC codes.The first research topic is the error floor behavior of FL-GLDPC and FL-DGLDPC codes.In GLDPC and DGLDPC codes, the existence of generalized CNs and VNs requires more careful management of edge connections for achieving low error floors.In this work the approximate cycle extrinsic message degree (ACE), a cycle parameter used in LDPC codes that reflects the resilience of the cycle to the noise, is generalized to extended ACE (eACE) for GLDPC and DGLDPC codes.By integrating eACE into the progressive-edge-growth (PEG) algorithm and adding more constraints on the connectivity of nodes, the error floor performance of GLDPC and DGLDPC codes can be greatly improved.In practice, designing codes capable of fast encoding is an important issue.Efficiently-encodable (EE) GLDPC and DGLDPC codes are studied and a self-consistent unified framework is proposed for constructing EE iteratively-decodable block codes, including EE-LDPC, EE-GLDPC and EE-DGLDPC codes.The competing nature of the fast encoding speed and the low error floor performance iv Abstract v is analyzed and verified via various simulation results.EE codes are systematic and suitable for puncturing; therefore, they are good candidates for rate-compatible (RC) data transmissions.Based on the characteristics of GLDPC and DGLDPC codes, we devise several systematic puncturing algorithms to provide good puncturing patterns for them.It is shown that we can significantly improve the code performance compared with random puncturing.Over a wide range of code rates, RC EE-GLDPC and EE-DGLDPC codes show advantages over EE-LDPC codes, especially at high signal-to-noise ratios (SNRs).
This dissertation studies methods to achieve reliable communication over unreliable channels. Iterative decoding algorithms for low-density parity-check (LDPC) codes and generalized LDPC (GLDPC) codes are analyzed.\n\nA new class of error-correcting codes to enhance the reliability of the communication for high-speed systems, such as optical communication systems, is proposed. The class of spatially-coupled GLDPC codes is studied, and a new iterative hard- decision decoding (HDD) algorithm for GLDPC codes is introduced. The main result is that the minimal redundancy allowed by Shannon’s Channel Coding Theorem can be achieved by using the new iterative HDD algorithm with spatially-coupled GLDPC codes. A variety of low-density parity-check (LDPC) ensembles have now been observed to approach capacity with iterative decoding. However, all of them use soft (i.e., non-binary) messages and a posteriori probability (APP) decoding of their component codes. To the best of our knowledge, this is the first system that can approach the channel capacity using iterative HDD.\n\nThe optimality of a codeword returned by the weighted min-sum (WMS) algorithm, an iterative decoding algorithm which is widely used in practice, is studied as well. The attenuated max-product (AttMP) decoding and weighted min-sum (WMS) decoding for LDPC codes are analyzed. Applying the max-product (and belief- propagation) algorithms to loopy graphs are now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence and/or the optimality of converged solutions. This work presents an analysis of both AttMP decoding and WMS decoding for LDPC codes which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution.