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With the increasing importance of digital communications and data storage, there is a need for research in the area of coding theory and channel modelling to design codes for channels that are power limited and/or bandwidth limited. Typical examples of such channels are found in cellular communications (GSM, UMTS), fixed and mobile broadband wireless access (WiMax) and magnetic storage media. Our motivation to write this book is twofold. Firstly, it is intended to provide PhD students and researchers in engineering with a sound background in binary and nonbinary error-correcting codes
Covering different classes of block and convolutional codes, band-width efficient coded modulation techniques and spatial-temporal diversity. Secondly, it is also intended to be suitable as a reference text for postgraduate students enrolled on Master’s-level degree courses and projects in channel coding. Chapter 1 of this book introduces the fundamentals of information theory and concepts, followed by an explanation of the properties of fading channels and descriptions of different channel models for fixed wireless access, mobile communications systems and magnetic storage.
In Chapter 2, basic mathematical concepts are presented in order to explain nonbinary error-correction coding techniques, such as Groups, Rings and Fields and their properties, in order to understand the construction of ring trellis coded modulation (ring-TCM), ring block coded modulation (ring-BCM), Reed–Solomon codes and algebraic–geometric codes. Binary and non-binary block codes and their application to wireless communications and data storage are discussed in Chapter 3, covering the construction of binary and non-binary Bose–Chaudhuri–Hocquengem (BCH) codes, Reed–Solomon codes and binary and non-binary BCM codes.
Chapter 4 introduces the construction methods of algebraic–geometric (AG) codes, which require an understanding of algebraic geometry. The coding parameters of AG codes are compared with Reed–Solomon codes and their performance and complexity are evaluated. Simulation results showing the performance of AG and Reed–Solomon codes are presented on fading channels and on magnetic storage channels. Chapter 5 presents an alternative decoding algorithm known as list decoding, which is applied to Reed–Solomon codes and AG codes.
Hard-decision list decoding for these codes is introduced first, using the Guruswami–Sudan algorithm. This is then followed by soft-decision list decoding, explaining the Kotter-Vardy algorithm for Reed–Solomon codes and modifying it for AG codes. The performance and complexity of the list decoding algorithms for both Reed–Solomon codes and AG codes are evaluated. Simulation results for hard- and soft-decision list decoding of these codes on AWGN and fading channels are presented and it is shown that coding gains over conventional hard-decision decoding can be achieved. A more recent coding scheme known as the low-density parity check (LDPC) code is introduced in Chapter 6.
This is an important class of block code capable of near-Shannon limit performance, constructed from a sparse parity check matrix. We begin by explaining the construction and decoding of binary LDPC codes and extend these principles to non-binary LDPC codes. Finally, the reduction of the decoding complexity of non-binary LDPC codes using fast Fourier transforms (FFTs) is explained in detail, with examples. Chapter 7 begins with an explanation of convolutional codes and shows how they can be combined with digital modulation to create a class of bandwidth-efficient codes called TCM codes. The construction of TCM codes defined over rings of integers, known as ring-TCM codes, is explained, and the design of good ring-TCM codes using a Genetic algorithm is presented.
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