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.