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Cambridge University Press eBooks

Conclusion and Remarks

Juane Li, Shu Lin, Khaled Abdel-Ghaffar, W.E. Ryan, Daniel J. Costello

Published Dec 1, 2016• 0 citations
Publisher

Abstract

In this book, we unified several major types of LDPC code constructions using a single framework, namely, the SP-construction. Under this framework, the constructions of all these types of LDPC codes can be viewed as basically algebraic. In general, algebraically constructed LDPC codes have good overall performance in terms of both waterfall and error-floor performance, as well as a fast rate of decoding convergence. The unification of all these constructions may lead to better designs, and construction of high performing and more easily implementable codes for applications in next generations of communication and data storage systems. This book consists of seven parts. In the first part (Chapter 4), we gave an algebraic interpretation of the protograph-based (PTG-based) construction of LDPC codes and presented a simple and novel algebraic method for constructing PTG-LDPC codes. The proposed algebraic method is equivalent to the graphical copy-and-permute operation and uses a simple matrix decomposition-and-replacement process to construct PTG-LDPC codes. The algebraic method is based on decomposition of a small base matrix and is very flexible in code construction. The constructed codes perform well as supported by examples and simulation results. However, how to design decomposition base matrices so that the resultant PTG-LDPC codes can perform close to their decoding thresholds is still an unresolved problem which needs further investigation. In the second part of the book (Chapters 5 and 6), we re-interpreted the SP-construction of LDPC codes, one of the earliest methods for algebraic construction of LDPC codes, in a broader context, from both algebraic and graph-theoretic perspectives. We showed that the PTG-LDPC code construction is a special case of SP-construction. New constructions of RC-constrained base matrices and PW-RC-constrained replacement sets of matrices for the SP-construction of LDPC codes were presented. In Chapter 5, we showed that two major ensembles of SP-LDPC codes for a given rate can be formed. One ensemble is equivalent to the ensemble of PTG-LDPC codes. In forming this ensemble, the member matrices in the replacement set R for the SP-construction of LDPC codes are regular. Since this ensemble is equivalent to the ensemble of PTG-LDPC codes, the SP-LDPC codes in this ensemble have good asymptotic performance and structural properties. For the other ensemble, the member matrices in the replacement set R for the SP-construction of LDPC codes are not regular.

DOI
10.1017/9781316780152.013

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In this book, we unified several major types of LDPC code constructions using a single framework, namely, the SP-construction. Under this framework, the constructions of all these types of LDPC codes can be viewed as basically algebraic. In general, algebraically constructed LDPC codes have good overall performance in terms of both waterfall and error-floor performance, as well as a fast rate of decoding convergence. The unification of all these constructions may lead to better designs, and construction of high performing and more easily implementable codes for applications in next generations of communication and data storage systems. This book consists of seven parts. In the first part (Chapter 4), we gave an algebraic interpretation of the protograph-based (PTG-based) construction of LDPC codes and presented a simple and novel algebraic method for constructing PTG-LDPC codes. The proposed algebraic method is equivalent to the graphical copy-and-permute operation and uses a simple matrix decomposition-and-replacement process to construct PTG-LDPC codes. The algebraic method is based on decomposition of a small base matrix and is very flexible in code construction. The constructed codes perform well as supported by examples and simulation results. However, how to design decomposition base matrices so that the resultant PTG-LDPC codes can perform close to their decoding thresholds is still an unresolved problem which needs further investigation. In the second part of the book (Chapters 5 and 6), we re-interpreted the SP-construction of LDPC codes, one of the earliest methods for algebraic construction of LDPC codes, in a broader context, from both algebraic and graph-theoretic perspectives. We showed that the PTG-LDPC code construction is a special case of SP-construction. New constructions of RC-constrained base matrices and PW-RC-constrained replacement sets of matrices for the SP-construction of LDPC codes were presented. In Chapter 5, we showed that two major ensembles of SP-LDPC codes for a given rate can be formed. One ensemble is equivalent to the ensemble of PTG-LDPC codes. In forming this ensemble, the member matrices in the replacement set R for the SP-construction of LDPC codes are regular. Since this ensemble is equivalent to the ensemble of PTG-LDPC codes, the SP-LDPC codes in this ensemble have good asymptotic performance and structural properties. For the other ensemble, the member matrices in the replacement set R for the SP-construction of LDPC codes are not regular.
This paper unveils a strong connection between two major constructions of LDPC codes, namely the algebraic-based and the protograph-based constructions. It is shown that, from a graph-theoretic point of view, an algebraic LDPC code whose parity-check matrix is an array of submatrices of the same size over a finite field is a protograph LDPC code. Conversely, from a matrix-theoretic point of view, since the parity-check matrix of a protograph code can be arranged as an array of submatrices of the same size over a finite field and its base graph (or base matrix) can be constructed algebraically, a protograph LDPC code is an algebraic LDPC code. These two major approaches have their advantages and disadvantages in code construction. Unification of these two approaches may lead to better designs and constructions of LDPC codes to achieve good overall performance in terms of error performance in waterfall region, error-floor location and rate of decoding convergence. This paper is the first part of a series of two parts, Part-I and Part-II. Part-I investigates only the binary LDPC codes constructed by the superposition and the protograph-based methods. Part-II explores nonbinary LDPC codes from both superposition and protograph points of view. Also included in Part II are specific superposition constructions of both binary and nonbinary quasi-cyclic LDPC codes.
This paper focuses on the construction of protograph low-density parity-check (PTG-LDPC) codes. We first give a necessary condition for a PTG-LDPC code to avoid 4-cycles. Then, we propose a new method for constructing PTG-LDPC codes based on the decomposition of a base matrix and the algebraic construction of an exponent matrix. The proposed method owns a lower design complexity compared with the conventional graph-based methods. Numerical results show that the proposed codes perform well, and have no error floor down to a block error rate of 10-4.
Dans ce travail nous nous intéressons aux problèmes liés à l'amélioration des performances d'erreur des codes LDPC. Cette amélioration peut être faite pendant la construction des codes LDPC ou pendant leur processus de décodage. Nous concentrons notre attention vers l'amélioration du décodage. Nous proposons un algorithme de décodage itératif log-SPA modifié, qui minimise l'erreur du décodage pour les mots-codes qui ne peuvent pas être décodés par l'algorithme SPA connu. L'algorithme proposé est basé sur une dépendance entre le poids d'un syndrome et le poids de la séquence d'erreurs généré au cours des itérations du décodage. Nous analysons plus en détail cette dépendance et l'effet du décodage log-SPA moyenne sur les poids des vecteurs d'erreur et les poids du syndrome, et ce, sur les ensembles de piégeage. Concernant la construction des codes LDPC, on s'intéresse aux différents paramètres qui posent des problèmes dans la performance d'erreur : au plancher d'erreur, au périmètre du graphe de Tanner, à la distance minimale du code, aux ensembles de piégeage et aux ensembles d'arrêt. On présente différentes méthodes appliquées dans la construction des différents types de codes, pour éviter l'apparition des structures non-souhaitables dans le code. Nous faisons une comparaison entre la performance des codes pseudo-aléatoires et les codes structurés, basés sur les geometries finies euclidienne et géométrique. On présente aussi différentes méthodes de construction des codes LDPC quasi-cycliques, qui peuvent éviter la formation de certains types d'ensembles de piégeage. d'ensembles d'arrêt ainsi que les cycles courts dans le graphe de Tanner. Les méthodes proposées donnent de très bas planchers d'erreur et une faible complexité du codage.
In this paper, a new systematic construction of optimized protograph generalized low-density parity-check codes with good decoding threshold and low error floor is proposed. First, a typical code graph is generated by combining an accumulate-repeat-accumulate seed protograph with a Tanner graph extension of a simple linear block node. Subsequently, it is analyzed and optimized theoretically, especially with the puncture mechanism, by a modified extrinsic information transfer chart and an asymptotic weight distribution. The generated graph is then extended to a base matrix by a copy-and-permute procedure, accompanied with the matrix split, which is optimized by a progressive edge growth for good randomness and girth property. Finally, the proposed code matrix is created by replacing “1”in the above base matrix with square circulant sub-matrices, the offsets of which are searched by a quasi-cyclic (QC)-oriented modified approximate cycle extrinsic message degree algorithm to improve the cycle relationship, especially for the compound cycles. Simulation results show that the codes exhibit excellent performance in both error floor and waterfall region on an additive white Gaussian noise channel. Moreover, they are also characterized by faster encoding due to the QC structure as well as lower decoding complexity and less latency, which make them a natural fit for new-generation power constrained wireless communications.