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:



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

(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 $d = d_1 \cdot d_2 = 3 \cdot 3 \ \underline{= 9}$.