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.