Linear Codes

Abstract

In Chapter 6 we considered several examples of linear codes, and in Lemma 6.8 we have already seen one advantage of dealing with them, namely that calculating the minimum distance of a linear code is easier than for general codes. In this chapter we will study linear codes in greater detail, noting other advantages to be obtained by applying elementary linear algebra and matrix theory, including an even simpler method for calculating the minimum distance. The theoretical background required includes such topics as linear independence, dimension, and row and column operations. These are normally covered in any first-year university linear algebra course; although such courses often restrict attention to vector spaces and matrices over the fields of real or complex numbers, all the important results and techniques we need extend in the obvious way to arbitrary fields, including finite fields. Throughout this chapter, we will assume that the alphabet F is the finite field F _q of order q, for some primepower q=p ^e.