Error control codes protect the accuracy of data in modern information systems, including computing, communication, and storage systems. Low-density parity-check (LDPC) codes and their relatives represent the state of the art in error control coding and are renowned for their ability to perform close to the theoretical limits. This book presents recent results on various LDPC code designs, making strong connections between two prominent design approaches, the algebraic-based and the graph-theoretic-based constructions. New codes and code construction techniques are presented.
Most methods for constructing LDPC codes can be classified into two general categories, the algebraic-based and the graph-theoretic-based constructions. The two best-known graph-theoretic-based construction methods are the progressive edge-growth (PEG) and the protograph-based (PTG-based) methods, devised in 2001 and 2003, respectively. Both of these techniques involve computer-aided design. One of the earliest algebraic-based methods for constructing LDPC codes is the superposition (SP) construction, proposed in 2002. In this book, the algebraic-based construction method is re-interpreted from both the algebraic and the graph-theoretic perspectives. From the algebraic point of view, it is shown that the SP-construction of LDPC codes includes, as special cases, most of the major algebraic construction methods developed since 2002. From the graph-theoretic point of view, it is shown that the SP-construction also includes the PTG-based construction as a special case. Based on this PTG/SP connection, an algebraic method is developed here to construct PTG-based LDPC codes.
There are advantages to putting the algebraic-based and the PTG-based constructions into a single framework, the SP framework. One advantage is that SP descriptions of codes tend to be relatively compact, enabling simple code specifications in standards and textbooks. Another advantage to studying LDPC codes under the SP framework is that students and practitioners need only learn a single code design approach rather than the myriad approaches that exist in the published literature.
Both binary and nonbinary code constructions will be presented under the SP framework. The SP-construction also leads to a new class of LDPC codes with a doubly quasi-cyclic (QC) structure as well as algebraic methods for constructing spatially and globally coupled LDPC codes. The globally coupled codes will be shown to possess a highly effective burst-erasure correction capability.