Error Correction Coding, 2nd Edition

Chapter 2Groups and Vector Spaces

2.1 Introduction

Linear block codes form a group and a vector space. Hence, the study of the properties of this class of codes benefits from a formal introduction to these concepts. The codes, in turn, reinforce the concepts of groups and subgroups that are valuable in the remainder of our study.

Our study of groups leads us to cyclic groups, subgroups, cosets, and factor groups. These concepts, important in their own right, also build insight in understanding the construction of extension fields which are essential for some coding algorithms to be developed.

2.2 Groups

A group formalizes some of the basic rules of arithmetic necessary for cancellation and solution of some simple algebraic equations.

Definition 2.1

A binary operation on a set is a rule that assigns to each ordered pair of elements of the set some element of the set. (Since the operation returns an element in the set, this is actually defined as closed binary operation. We assume that all binary operations are closed.)

Example 2.2

On the set of positive integers, we can define a binary operation by .

Example 2.3

On the set of real numbers, we can define ...