Since a code is a finite dimensional vector space over a finite field, it only has finitely many elements. To represent a code in a computer one may of course store all the elements. But there must be a simpler way of representing a code than by listing all of its elements, right? Yes, there is and this motivates the following definition.

A code is often represented by simply giving a generator matrix. To compute a generator matrix of a given code of length , first determine a basis for the code as a vector space over , then put these basis vectors into a matrix, where .

*(a) its sum with another row, or*

*(b) its scalar multiple with any non-zero element of ,*

*is called an elementary row operation. *

*If is any matrix with entries in a field then*

*(a) swapping any two columns, or*

*(b) replacing any column of by its scalar multiple with any non-zero element of ,*

*is called a simple column operation.*

**proof**: The rows of still form a basis for the vector space over .

*However, if we swap the first two columns of , to get*

*These are called generalized (or shortened or extended) Reed-Solomon codes. (As we shall see later, in the context of cyclic codes, Reed-Solomon codes have .) Note the parameters satisfy the Singleton bound, so these are MDS codes.*

*The generator matrix for is of the form .*

The **encoding matrix** of a linear code of dinension is an matrix whose image is . To compute the encoding matrix from the generator matrix is easy.

The proof of this is left as an exercise.

Many general properties about a linear code may be reduced to a corresponding property of its generator matrix. Can one determine the minimum distance of a code from its generator matrix? The following result answers this.

**proof**: By definition of , there is a code word in having exactly non-zero entries. Thus by definition of , there are columns of which are linearly dependent. If there were columns of which are linearly independent then there would be (by definition of ) a code word in having exactly non-zero entries. This would contradict the definition of .

David Joyner 2007-09-03