Majority decoding is a method of decoding by voting that is simple to implement and is extremely fast. However, only a small class of codes can be majority decoded, and usually these codes are not as good as other codes. Because code performance is usually more important than decoder simplicity, majority decoding is not important in most applications. Nevertheless, the theory of majority-decodable codes provides another well-developed view of the subject of error-control codes. The topic of majority decoding has connections with combinatorics and with the study of finite geometries.

Most known codes that are suitable for majority decoding are cyclic codes or extended cyclic codes. For these codes, the majority decoders can always be implemented as Meggitt decoders and characterized by an especially simple logic tree for examining the syndromes. Thus one can take the pragmatic view and define majority-decodable codes as those cyclic codes for which the Meggitt decoder can be put in a standard simple form. But in order to find these codes, we must travel a winding road.

Reed–Muller codes

Reed–Muller codes are a class of linear codes over GF(2) that are easy to describe and have an elegant structure. They are an example of a code that can be decoded by majority decoding. For these reasons, the Reed–Muller codes are important even though, with some exceptions, their minimum distances are not noteworthy. Each Reed–Muller code can be punctured to give one of the cyclic Reed–Muller codes that were studied in Section 6.7.

Cyclic codes are a subclass of the class of linear codes obtained by imposing an additional strong structural requirement on the codes. Because of this structure, the search for error-control codes has been most successful within the class of cyclic codes. Here the theory of Galois fields has been used as a mathematical searchlight to spot the good codes. Outside the class of cyclic codes, the theory of Galois fields casts a dimmer light. Most of what has been accomplished builds on the ideas developed for cyclic codes.

The cyclic property in itself is not important and cyclic codes do not necessarily have a large minimum distance. Cyclic codes are introduced because their structure is closely related to the strong structure of the Galois fields. This is significant because the underlying Galois-field description of a cyclic code leads to encoding and decoding procedures that are algorithmic and computationally efficient. Algorithmic techniques have important practical applications, in contrast to the tabular decoding techniques that are used for arbitrary linear codes.

This chapter gives a leisurely introduction to cyclic codes as a special class of linear codes. An alternative approach to the topic of cyclic codes, based on the Fourier transform, is given in Chapter 6.

The search for good data-transmission codes has relied, to a large extent, on the powerful and beautiful structures of modern algebra. Many important codes, based on the mathematical structures known as Galois fields, have been discovered. Further, this algebraic framework provides the necessary tools with which to design encoders and decoders. This chapter and Chapter 4 are devoted to developing those topics in algebra that are significant to the theory of data-transmission codes. The treatment is rigorous, but it is limited to material that will be used in later chapters.

Fields of characteristic two

Real numbers form a familiar set of mathematical objects that can be added, subtracted, multiplied, and divided. Similarly, complex numbers form a set of objects that can be added, subtracted, multiplied, and divided. Both of these arithmetic systems are of fundamental importance in engineering disciplines. We will need to develop other, less familiar, arithmetic systems that are useful in the study of data-transmission codes. These new arithmetic systems consist of sets together with operations on the elements of the sets. We shall call the operations “addition,” “subtraction,” “multiplication,” and “division,” although they need not be the same operations as those of elementary arithmetic.

Modern algebraic theory classifies the many arithmetic systems it studies according to their mathematical strength. Later in this chapter, these classifications will be defined formally. For now, we have the following loose definitions.

1. Abelian group. A set of mathematical objects that can be “added” and “subtracted.”

2. Ring. A set of mathematical objects that can be “added,” “subtracted,” and “multiplied.”

3. […]

To encode an infinite stream of data symbols with a block code, the datastream is broken into blocks, each of k data symbols, called datawords. The block code encodes each block of k data symbols into a block of n code symbols, called a codeword. The codewords are concatenated to form an infinite stream of code symbols.

In contrast to a block code is a trellis code. A trellis code also encodes a stream of data symbols into a stream of code symbols. A trellis code divides the datastream into blocks of length k, called dataframes, which are usually much smaller and are encoded into blocks of length n, called codeframes. In both cases, block codes and trellis codes, the datastream is broken into a sequence of blocks or frames, as is the codestream. When using a block code, a single block of the codestream depends only on a single block of the datastream, whereas when using a trellis code, a single frame of the codestream depends on multiple frames of the datastream.

The most important trellis codes are those known as convolutional codes. Convolutional codes are trellis codes that satisfy certain additional linearity and time-invariance properties. Although we introduce the general notion of a trellis code, we will be concerned mostly with the special class of convolutional codes.