Block codes - Codes and Turbo Codes - page 111

Matrix G with k rows and n columns, having its elements g jl ∈ F 2 is called

a generator matrix of the code C ( n, k ) . It associates the codeword c with the
block of data d by the matrix relation:

c = dG (4.5)

The generator matrix of a block code is not unique. Indeed, by permuting
the vectors of the base ( e 0 ,..., e l ,..., e n− 1 ) or of the base ( e 0 ,..., e j ,...,
e k− 1 ) , we obtain a new generator matrix G whose columns or rows have also
been permuted. Of course, the permutation of the columns or the rows of the
generator matrix always produces the same set of codewords; what changes is
the association between the codewords and the k -uplets of data.
Note that the rows of the generator matrix of a linear block code are in-
dependent codewords, and that they make up a base of the vector subspace
generated by the code. The generator matrix of a linear block code is therefore
of rank k . A direct consequence is that any family made up of k independent
codewords can be used to define a generator matrix of the code considered.

Example 4.1

Let us consider a linear block code called the parity check code denoted
C ( n, k ) ,with k =2 and n = k +1=3 (for a parity check code, the sum of the
symbols of a codeword is equal to zero). We have four codewords:

Dataword Codeword

00 000

01 011

10 101

11 110

To write a generator matrix of this code, let us consider, for example, the canon-
ical base of F 2 :

e 0 = 10 , e 1 = 01
and the canonical base of F 2 :
e 0 = 100 , e 1 = 010 , e 2 = 001

We can write:

g ( e 0 ) = [101] = 1 . e 0 +0 . e 1 +1 . e 2
g ( e 1 ) = [011] = 0 . e 0 +1 . e 1 +1 . e 2
A generator matrix of the parity check code is therefore equal to :
G = 101
011