Block codes - Codes and Turbo Codes - page 113

of the vector subspace made up of 2 k words of the code C ( n, k ) . It results that
any word c of code C ( n, k ) is orthogonal to the rows of the generator matrix H
of its dual code

cH T = 0 (4.8)

where T indicates the transposition.
A vector y belonging to F 2 is therefore a codeword of code C ( n, k ) if, and
only if, it is orthogonal to the codewords of its dual code, that is, if:

yH T = 0

The decoder of a code C ( n, k ) can use this property to verify that the word
received is a codeword and thus to detect the presence of errors. That is why
matrix H is called the parity check matrix of code C ( n, k ) .
It is easy to see that the matrices G and H are orthogonal ( GH T = 0 ) .
Hence, when the code is systematic and its generator matrix is of the form
G =[ I k P ] ,wehave:

H =[ P T I n−k ] (4.9)

4.1.3 Minimum distance
Before recalling what the minimum distance of a linear block code is, let return
to the notion of Hamming distance that measures the difference between two
codewords. The Hamming distance, denoted d H , is equal to the number of
places where the two codewords have different symbols.
We can also define the Hamming weight, denoted w H ,ofacodewordasthe
number of non-null symbols of this codeword. Thus, the Hamming distance
between two codewords is also equal to the weight of their sum.

Example 4.2

Let there be two words u = [1101001] and v = [0101101] . The Hamming
distance between u and v is 2 .Theirsum u + v = [1000100] has a Hamming
weight 2 .

The minimum distance d min of a block code is equal to the smallest Hamming
distance between its codewords.

d H ( c , c ) , c , c ∈ d min =min

c = c

∀ C ( n, k ) (4.10)

Taking into account the fact that the distance between two codewords is equal
to the weight of their sum, the minimum distance of a block code is also equal
to the minimum weight of its non-null codewords.