Difference between revisions of "Aufgaben:Exercise 4.6Z: Basics of Product Codes"

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{{quiz-Header|Buchseite=Channel_Coding/The_Basics_of_Product_Codes}}
 
{{quiz-Header|Buchseite=Channel_Coding/The_Basics_of_Product_Codes}}
  
[[File:P_ID3002__KC_Z_4_6_v3.png|right|frame|Generator matrices of the component codes]]
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[[File:P_ID3002__KC_Z_4_6_v3.png|right|frame|Generator matrices of <br>the component codes]]
We consider here a product code according to the description in section&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code|"Basic structure of a Product Code"]]. The two component codes&nbsp; $\mathcal{C}_1$&nbsp; and&nbsp; $\mathcal{C}_2$&nbsp; are defined by the generator matrices&nbsp; $\mathbf{G}_1$&nbsp; and&nbsp; $\mathbf{G}_2$&nbsp; given on the right.
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We consider here a product code according to the description in section&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code|"Basic structure of a Product Code"]].&nbsp; The two component codes&nbsp; $\mathcal{C}_1$&nbsp; and&nbsp; $\mathcal{C}_2$&nbsp; are defined by the generator matrices&nbsp; $\mathbf{G}_1$&nbsp; and&nbsp; $\mathbf{G}_2$&nbsp; given on the right.
 
 
 
 
  
  
  
  
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<u>Hints:</u>
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*This exercise belongs to the chapter&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes|"Basics of a Product Code"]].
  
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*Reference is made  to the section&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_Product_Code|"Basic structure of a product code"]].
  
Hints:
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*The two component codes are also covered in the&nbsp; [[Aufgaben:Aufgabe_4.6:_Produktcode–Generierung|$\text{Exercise 4.6}$]]&nbsp;.
*This exercise belongs to the chapter&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes|"Basics of a Product Code"]].
 
*Reference is made in particular to the section&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_Product_Code|"Basic structure of a product code"]].
 
*The two component codes are also covered in the&nbsp; [[Aufgaben:Aufgabe_4.6:_Produktcode–Generierung|"Exercise 4.6"]]&nbsp;.
 
  
  

Revision as of 18:19, 6 December 2022

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:



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