This is the chapter where the reader first approaches the world of networks. The preliminary step to take consists in providing a succinct introduction to the analysis of the signals that, either in the electrical or in the optical domain, carry data information across networks.

Also, before throwing onto the floor the first Lego bricks and the instructions to let the reader build the composite view of modern computer networks, a review of the “fabric” of these bricks is needed. A light excursus is thus provided, to gain a basic acquaintance with copper wires, fiber optics and the radio channel, each transmission medium being employed in specific networking contexts.

Network classification comes next: it is a useful exercise to gain familiarity with the main ideas and the terminology peculiar to the world of networks. This is followed by the introduction of the crucial notion of delay, as well as by a miscellanea of concepts that cross different areas, spanning from sources of network traffic to service classification, from performance metrics to quality of service.

Following a pattern that will shape the exposition throughout the entire book, a direct experience is proposed and commented at the end of the chapter, to let the reader confront the real world via a first-hand adventure.

Signals: time and frequency analysis

Indeed, our aim is to talk about networks. Before doing so, however, we have to recall the fundamentals of signal analysis, to understand what type of information we are moving around.

The previous chapter examined code properties responsible for the floor region (high-SNR region) of LDPC and turbo codes. Specifically, the emphasis was on the computation of various weight enumerators for LDPC and turbo code ensembles because a poor weight spectrum leads to poor performance in a code's floor region both for iterative decoders and for maximum-likelihood decoders. That chapter also introduced ensemble enumerators for trapping sets and stopping sets, both of which can lead to poor floor performance in iterative decoders. In this chapter we examine the iterative decoding performance of LDPC and turbo code ensembles in the complementary low-SNR region, the “waterfall” region. We show that the iterative decoding of long LDPC and turbo codes displays a threshold effect in which communication is reliable beyond this threshold and unreliable below it. The threshold is a function of code ensemble properties and the tools introduced in this chapter allow the designer to predict the decoding threshold and its gap from Shannon's limit. The ensemble properties for LDPC codes are the degree distributions which are typically the design targets for the code-design techniques presented in subsequent chapters. The ensemble properties for turbo codes are the selected constituent codes. Our development borrows heavily from the references listed at the end of the chapter and our focus is on the binary-input AWGN channel. The references and the problems consider other channels.

Density Evolution for Regular LDPC Codes

We first summarize the main results of [1] with regard to iterative (message passing) decoding of long, regular LDPC codes.

There are two structurally different types of codes, block and convolutional codes. Both types of codes have been widely used for error control in communication and/or storage systems. Block codes can be divided into two categories, linear and nonlinear block codes. Nonlinear block codes are never used in practical applications and not much investigated. This chapter gives an introduction to linear block codes. The coverage of this chapter includes (1) fundamental concepts and structures of linear block codes; (2) specifications of these codes in terms of their generator and parity-check matrices; (3) their error-correction capabilities; (4) several important classes of linear block codes; and (5) encoding of two special classes of linear block codes, namely cyclic and quasi-cyclic codes. In our presentation, no proofs or derivations are provided; only descriptions and constructions of codes are given. We will begin our introduction with linear block codes with symbols from the binary field GF(2). Linear block codes over nonbinary fields will be given at the end of this chapter.

There are many excellent texts on the subject of error-control coding theory [1–15], which have extensive coverage of linear block codes. For in-depth study of linear block codes, readers are referred to these texts.

Introduction to Linear Block Codes

We assume that the output of an information source is a continuous sequence of binary symbols over GF(2), called an information sequence.

This final chapter is dedicated to a brief overview of some of the most important solutions used by network operators, on the one hand, for providing access to their customers, on the other, as wide area network (WAN) technologies are adopted in the network core. The chapter starts with an overview of the xDSL family, one of the most popular access technologies adopted today. Then a historical perspective of WAN technologies is given, going from the X.25 and ISDN solutions to the more recent frame relay service and ATM protocol. The attempt made by IETF to join the flexibility of the connectionless IP world with the advantages of the connection-oriented ATM approach is, finally, the basis for introducing the principles of MPLS, a very popular technology currently used by operators to offer enhanced IP-based connectivity services to their customers, including quality of service guarantees and tunneling capabilities. Given the importance of the last topic, the chapter closes with a section dedicated to a practical example of MPLS router configuration.

The xDSL family

The xDSL acronym refers to a set of solutions and technologies developed for supporting the transmission of high speed data in existing copper access networks. It also supports data transmissions at long distances but at lower bit rates.

The ITU-T recommendation G.995.1 [91] provides an overview of the digital subscriber line family. In the following ADSL, with the recent ADSL2 and ADSL2+, HDSL, SHDSL, VDSL and VDSL2 will be presented.

In Chapter 6, the main functions and architectures of routers have been presented and discussed. Routers are the basic active devices employed in networks and their fundamental task is to forward packets from node to node toward their final destination, taking the most adequate decisions concerning the network path to follow [65]. More specifically, routers perform three main functions: routing table creation and update, table look-up for forwarding decision and physical datagram forwarding from input ports to output ports.

The first of these functions is called routing and consists of running algorithms and implementing protocols suitable to the specific routing problem and network topology size [81].On the one hand, routing algorithms are executed to create and maintain routing tables, once sufficient information about the network topology has been gathered. On the other hand, routing protocols are employed between routers to exchange the network topology information necessary for executing the algorithms.

This chapter illustrates the basic principles of routing, describing the most important algorithms and protocols. The hierarchical approach to the routing problem adopted by the Internet is also discussed, including a brief overview of the most popular IP routing protocols, namely RIP and OSPF. A couple of practical examples of IP routing configurations conclude the chapter.

Routing algorithms

Routing algorithms are characterized by the way the routing table is created and possibly updated. Generally speaking, such an algorithm should be correct, fair, stable, optimal, fast, simple, and adaptive to network topology changes and to traffic conditions. It is evidently difficult to meet all these requirements with just a single, ideal algorithm, so any real routing algorithm usually provides a trade-off on some of them.

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.