Although a great deal of research effort has been expended on the design, construction, encoding, decoding, performance analysis, and applications of binary LDPC codes in communication and storage systems, very little has been done on nonbinary LDPC codes in these respects. The first study of nonbinary LDPC codes was conducted by Davey and MacKay in 1998. In their paper, they generalized the SPA for decoding binary LDPC codes to decode q-ary LDPC codes, called QSPA. Later, in 2000, MacKay and Davey introduced a fast-Fourier-transform (FFT)-based QSPA to reduce the decoding computational complexity of QSPA. This decoding algorithm is referred to as FFT-QSPA. MacKay and Davey's work on FFT-QSPA was further improved by Barnault and Declercq in 2003 and Declercq and Fossorier in 2007. Significant works on the design, construction and analysis of nonbinary LDPC codes didn't appear until the middle of 2000. Results in these works are very encouraging. They show that nonbinary LDPC codes have a great potential to replace widely used RS codes in some applications in communication and storage systems. This chapter is devoted to nonbinary LDPC codes.

Just like binary LDPC codes, nonbinary LDPC codes can be classified into two major categories: (1) random-like nonbinary codes constructed by computer under certain design criteria or rules; and (2) structured nonbinary codes constructed on the basis of algebraic or combinatorial tools, such as finite fields and finite geometries.

This chapter presents some important elements of modern algebra and combinatorial mathematics, namely, finite fields, vector spaces, finite geometries, and graphs, that are needed in the presentation of the fundamentals of classical channel codes and various constructions of modern channel codes in the following chapters. The treatment of these mathematical elements is by no means rigorous and coverage is kept at an elementary level. There are many good text books on modern algebra, combinatorial mathematics, and graph theory that provide rigorous treatment and in-depth coverage of finite fields, vector spaces, finite geometries, and graphs. Some of these texts are listed at the end of this chapter.

Sets and Binary Operations

A set is a collection of certain objects, commonly called the elements of the set. A set and its elements will often be denoted by letters of an alphabet. Commonly, a set is represented by a capital letter and its elements are represented by lower-case letters (with or without subscripts). For example, X = {x1, x2, x3, x4, x5} is a set with five elements, x1, x2, x3, x4, and x5. A set S with a finite number of elements is called a finite set; otherwise, it is called an infinite set. In error-control coding theory, we mostly deal with finite sets.

Combinatorial designs [1–8] form an important branch in combinatorial mathematics. In the late 1950s and during the 1960s, special classes of combinatorial designs, such as balanced incomplete block designs, were used to construct error correcting codes, especial majority-logic-decodable codes. More recently, combinatorial designs were successfully used to construct structured LDPC codes [9–12]. LDPC codes of practical lengths constructed from several classes of combinatorial designs were shown to perform very well over the binary-input AWGN channel with iterative decoding.

Graphs form another important branch in combinatorial mathematics. They were also used to construct error-correcting codes in the early 1960s, but not very successfully. Only a few small classes of majority-logic-decodable codes were constructed. However, since the rediscovery of LDPC codes in the middle of the 1990s, graphs have become an important tool for constructing LDPC codes. One example is to use protographs for constructing iteratively decodable codes as described in Chapters 6 and 8.

This chapter presents several methods for constructing LDPC codes based on special types of combinatorial designs and graphs.

Balanced Incomplete Block Designs and LDPC Codes

Balanced incomplete block designs (BIBDs) form an important class of combinatorial designs. A special subclass of BIBDs can be used to construct RC-constrained matrices or arrays of CPMs from which LDPC codes can be constructed. This section gives a brief description of BIBDs. For an in-depth understanding of this subject, readers are referred to [1–8].

The topics this book touches lie within the networking field, an exciting area that in the last 20 years has experienced a stunning growth and gained an increasing popularity. Just as previous ages of modern society have been marked by technological advancements that significantly shaped them, from transistors to personal computers, our life is now molded by emails, our work and leisure time clocked by websites, our children daily accompanied by the Internet. What lies behind this boiling surface? What infrastructures and communication rules allow us to connect to the Internet from home through an ADSL connection? How does information travel on a high-speed backbone from our office to an overseas destination? Through a rigorous yet practical approach, the aim of this volume is to provide all the concepts needed for a thorough knowledge of networking technologies, as well as to breed the development of agile skills in modern network design.

After laying the common language foundations and the basic concepts and terminology within the field, the book is committed to a critical treatment of the technologies, protocols and devices adopted in contemporary networks. A special emphasis is placed on building an effective competence in all subject areas: hence, each topic is complemented by guided and commented practices, where proficiencies are challenged through real problems. The aim is to strengthen the abilities needed for present-day network design. The chapter structure reflects the authors’ pedagogical view: first build good foundations and gain expertise in each topic, then consolidate and confront real networking issues.

The title of this book, Channel Codes: Classical and Modern, was selected to reflect the fact that this book does indeed cover both classical and modern channel codes. It includes BCH codes, Reed–Solomon codes, convolutional codes, finite-geometry codes, turbo codes, low-density parity-check (LDPC) codes, and product codes. However, the title has a second interpretation. While the majority of this book is on LDPC codes, these can rightly be considered to be both classical (having been first discovered in 1961) and modern (having been rediscovered circa 1996). This is exemplified by David Forney's statement at his August 1999 IMA talk on codes on graphs, “It feels like the early days.” As another example of the classical/modern duality, finite-geometry codes were studied in the 1960s and thus are classical codes. However, they were rediscovered by Shu Lin et al. circa 2000 as a class of LDPC codes with very appealing features and are thus modern codes as well. The classical and modern incarnations of finite-geometry codes are distinguished by their decoders: one-step hard-decision decoding (classical) versus iterative soft-decision decoding (modern).

It has been 60 years since the publication in 1948 of Claude Shannon's celebrated A Mathematical Theory of Communication, which founded the fields of channel coding, source coding, and information theory. Shannon proved the existence of channel codes which ensure reliable communication provided that the information rate for a given code did not exceed the so-called capacity of the channel.

Weight enumerators or weight-enumerating functions are polynomials that represent in a compact way the input and/or output weight characteristics of the encoder for a code. The utility of weight enumerators is that they allow us to easily estimate, via the union bound, the performance of a maximum-likelihood (ML) decoder for the code. Given that turbo and LDPC codes employ suboptimal iterative decoders this may appear meaningless, but it is actually quite sensible for at least two reasons. One reason is that knowledge of ML-decoder performance bounds allows us to weed out weak codes. That is, if a code performs poorly for the ML decoder, we can expect it to perform poorly for an iterative decoder. Another reason for the ML-decoder approach is that the performance of an iterative decoder is generally approximately equal to that of its counterpart ML decoder, at least over a restricted range of SNRs. We saw this in Chapter 7 and we will see it again in Figure 8.5 in this chapter.

Another drawback to the union-bound/ML-decoder approach is that the bound diverges in the low-SNR region, which is precisely the region of interest when one is attempting to design codes that perform very close to the capacity limit. Thus, when attempting to design codes that are simultaneously effective in the floor region (high SNRs) and the waterfall region (low SNRs), the techniques introduced in this chapter should be supplemented with the techniques in the next chapter which are applicable to the low-SNR region.