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. […]