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 on page  "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 in particular to the page  "Basic structure of a product code".
• The two component codes are also covered in the  "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 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 codeword 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 codeword 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 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 on the "Basic structure of a Product Code" page.

• You can see the information block with $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,
• The codeword 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 $d = d_1 \cdot d_2 = 3 \cdot 3 \ \underline{= 9}$.