William E. Ryan

An unnoticed strong connection between algebraic-based and protograph-based LDPC codes, Part I: Binary case and interpretation

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.

LDPC Code Designs, Constructions, and Unification

Written by leading experts, this self-contained text provides systematic coverage of LDPC codes and their construction techniques, unifying both algebraic- and graph-based approaches into a single theoretical framework (the superposition construction). An algebraic method for constructing protograph LDPC codes is described, and entirely new codes and techniques are presented. These include a new class of LDPC codes with doubly quasi-cyclic structure, as well as algebraic methods for constructing spatially and globally coupled LDPC codes. Authoritative, yet written using accessible language, this text is essential reading for electrical engineers, computer scientists and mathematicians working in communications and information theory.

Globally coupled LDPC codes

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.

Conclusion and Remarks

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.