Exercise 4.6Z: Basics of Product Codes

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Generator matrices of
the component codes

We consider here a product code according to the description in section  "Basic structure of a Product Code".  The two component codes  $\mathcal{C}_1$  and  $\mathcal{C}_2$  are defined by the generator matrices  $\mathbf{G}_1$  and  $\mathbf{G}_2$  given on the right.

Hints:

• This exercise belongs to the chapter  "Basics of a Product Code".

• Reference is made to the section  "Basic structure of a product code".

• The two component codes are also covered in the  $\text{Exercise 4.6}$ .

Questions

1

What statements does the generator matrix  $\mathbf{G}_1$  allow about the code  $\mathcal{C}_1$?

	The code rate of  $\mathcal{C}_1$  is  $R_1 = 4/7$.
	The code  $\mathcal{C}_1$  is systematic.
	$\mathcal{C}_1$  is a truncated Hamming code.
	The minimum distance of this code is  $d_1 = 3$.

2

What statements does the generator matrix  $\mathbf{G}_2$  allow about the code  $\mathcal{C}_2$?

	The code rate of  $\mathcal{C}_2$  is  $R_2 = 4/7$.
	The code  $\mathcal{C}_2$  is systematic.
	$\mathcal{C}_2$  is a truncated Hamming code.
	The minimum distance of this code is  $d_2 = 3$.

3

Specify the parameters of the product code  $\mathcal{C} = \mathcal{C}_1 × \mathcal{C}_2$ .

$k \hspace{0.25cm} = \ $

$n \hspace{0.25cm} = \ $

$d \hspace{0.25cm} = \ $

$R \hspace{0.15cm} = \ $

Solution

Solution

(1)  Correct are the  statements 1, 2 and 4:

• The number of rows of the generator matrix  $\mathbf{G}_1$  indicates the length of the information block   ⇒   $k = 4$.

• The code word length is equal to the number of columns   ⇒   $n=4$   ⇒   Code rate $R = k/n = 4/7$.

• The code is systematic because the generator matrix  $\mathbf{G}_1$  starts with a  $4 × 4$  diagonal matrix.

• This is a  "normal"  Hamming code.

• For this,  with the code word length  $n$  and the number of check bits   ⇒   $m = n - k$,  the relation  $n = 2^m - 1$  holds.

• In the present case,  this is the  (normal)  Hamming code $\rm (7, \ 4, \ 3)$.

• The last parameter in this code label specifies the minimum distance   ⇒   $d_{\rm min} = 3$.

(2)  Correct are the  statements 2, 3 and 4:

• This is a truncated Hamming code with parameter  $n = 6, \ k = 3$  and  $d_{\rm min} = 3$,  also in systematic form.

• The code rate is  $R = 1/2$.

(3)  The basic structure of the product code is shown in the  "Basic structure of a Product Code"  section.

• You can see the information block with  $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,

• The code word length is the total number of all bits: $n = n_1 \cdot n_2 = 7 \cdot 6 \ \underline{= 42}$.

• The code rate is thus given by  $R = k/n = 12/42 = 2/7$.

• Or:   $R = R_1 \cdot R_2 = 4/7 \cdot 1/2 \ \underline{= 2/7} \approx 0.289$.

• The free distance is  $d = d_1 \cdot d_2 = 3 \cdot 3 \ \underline{= 9}$.