Error-Control Block Codes for Communications Engineers
An engineer s guide to the theory behind the common random-error-control block codes, and their applications in digital communications.
TABLE OF CONTENTS Error-Control Block Codes for Communications Engineers
Chapter 2: Introduction to Abstract Algebra
Overview
The search for good error-control codes has relied to a large extent on the structures of abstract algebra, such as the structures of groups, rings, Galois fields, and polynomials over Galois fields. The treatment on abstract algebra here is basically descriptive and necessary for understanding subsequent chapters. For further understanding of the material, readers should consult any textbooks on abstract algebra [1 5].
In the study of error-control codes, we will need to know the operations on the elements of a set which lead us to the concepts of groups, rings, and finite fields. For now, we have the following loose definitions:
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Group a set of elements that can be added and subtracted without leaving the set.
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Ring a set of elements that can be added, subtracted, and multiplied without leaving the set.
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Field a set of elements that can be added, subtracted, multiplied, and divided without leaving the set.
The addition, subtraction, multiplication and division are not the conventional arithmetic operations; these names are used because the operations resemble the conventional operations.
2.1 Groups
Let G be a set of elements.
Definition 2.1
A set G on which a binary operation * is defined is called a group. The binary operation * on G is a rule that assigns to each pair of elements a and b a uniquely defined third element c = a * b in G. We say that G is closed under * and the group must satisfy...
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