Aufgaben:Exercise 4.6Z: Basics of Product Codes - Revision history

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	[[Category:Channel Coding: Exercises\|^4.2 About the Product Codes^]]		[[Category:Channel Coding: Exercises\|^4.2 About the Product Codes^]]
	[[de:Aufgaben:4.~~6Z_Grundlagen_der_Produktcodes~~]]		[[de:Aufgaben:Aufgabe 4.6Z: Grundlagen der Produktcodes]]

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	[[Category:Channel Coding: Exercises\|^4.2 About the Product Codes^]]		[[Category:Channel Coding: Exercises\|^4.2 About the Product Codes^]]
			[[de:Aufgaben:4.6Z_Grundlagen_der_Produktcodes]]

Guenter at 16:29, 6 December 2022

2022-12-06T16:29:29Z

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  Correct are <u>statements 1, 2 and 4</u>:		'''(1)'''  Correct are the  <u>statements 1, 2 and 4</u>:
	* The number of rows of the generator matrix $\mathbf{G}_1$ indicates the length of the information block   ⇒   $k = 4$.		* 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 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 $4 &times 4$ diagonal matrix.
	*This is a "normal" Hamming code.		* The code is systematic because the generator matrix  $\mathbf{G}_1$  starts with a  $4 &times 4$  diagonal matrix.
	*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 $\rm (7, \ 4, \ 3)$.		*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   ⇒   $d_{\rm min} = 3$.		*The last parameter in this code label specifies the minimum distance   ⇒   $d_{\rm min} = 3$.


	'''(2)'''  Correct <u>statements 2, 3 and 4</u>:		'''(2)'''  Correct are the  <u>statements 2, 3 and 4</u>:
	*This is a truncated Hamming code with parameter $n = 6, \ k = 3$ and $d_{\rm min} = 3$, also in systematic form.		*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 $R = 1/2$.
			*The code rate is  $R = 1/2$.


	'''(3)'''  The basic structure of the product code is shown in the [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"Basic structure of a Product Code"]] section.		'''(3)'''  The basic structure of the product code is shown in the  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"Basic structure of a Product Code"]]  section.
	* You can see the information block with $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,		* 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 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 $R = k/n = 12/42 = 2/7$.
			*The code rate is thus given by  $R = k/n = 12/42 = 2/7$.

	*Or:   $R = R_1 \cdot R_2 = 4/7 \cdot 1/2 \ \underline{= 2/7} \approx 0.289$.		*Or:   $R = R_1 \cdot R_2 = 4/7 \cdot 1/2 \ \underline{= 2/7} \approx 0.289$.
	* The free distance is $d = d_1 \cdot d_2 = 3 \cdot 3 \ \underline{= 9}$.
			* The free distance is  $d = d_1 \cdot d_2 = 3 \cdot 3 \ \underline{= 9}$.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 16:21, 6 December 2022

2022-12-06T16:21:12Z

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	*Reference is made to the section  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_Product_Code\|"Basic structure of a product code"]].		*Reference is made to the section  [[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  [[Aufgaben:~~Aufgabe_4~~.6:~~_Produktcode–Generierung~~\|$\text{Exercise 4.6}$]] .		*The two component codes are also covered in the  [[Aufgaben:Exercise_4.6:_Product_Code_Generation\|$\text{Exercise 4.6}$]] .

Guenter at 16:19, 6 December 2022

2022-12-06T16:19:58Z

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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]]		[[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  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"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.		We consider here a product code according to the description in section  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"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.






			<u>Hints:</u>
			*This exercise belongs to the chapter  [[Channel_Coding/The_Basics_of_Product_Codes\|"Basics of a Product Code"]].

			*Reference is made to the section  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_Product_Code\|"Basic structure of a product code"]].

	~~Hints:~~		*The two component codes are also covered in the  [[Aufgaben:Aufgabe_4.6:_Produktcode–Generierung\|$\text{Exercise 4.6}$]] .
	*This exercise belongs to the chapter  [[Channel_Coding/The_Basics_of_Product_Codes\|"Basics of a Product Code"]].
	*Reference is made in particular to the section  [[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  [[Aufgaben:Aufgabe_4.6:_Produktcode–Generierung\|"Exercise 4.6"]] .

Hwang at 23:41, 12 November 2022

2022-11-12T23:41:38Z

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	[[File:P_ID3002__KC_Z_4_6_v3.png\|right\|frame\|Generator matrices of the component codes]]		[[File:P_ID3002__KC_Z_4_6_v3.png\|right\|frame\|Generator matrices of the component codes]]
	We consider here a product code according to the description ~~on page~~  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"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.		We consider here a product code according to the description in section  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"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.


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	Hints:		Hints:
	*This exercise belongs to the chapter  [[Channel_Coding/The_Basics_of_Product_Codes\|"Basics of a Product Code"]].		*This exercise belongs to the chapter  [[Channel_Coding/The_Basics_of_Product_Codes\|"Basics of a Product Code"]].
	*Reference is made in particular to the ~~page~~  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_Product_Code\|"Basic structure of a product code"]].		*Reference is made in particular to the section  [[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  [[Aufgaben:Aufgabe_4.6:_Produktcode–Generierung\|"Exercise 4.6"]] .		*The two component codes are also covered in the  [[Aufgaben:Aufgabe_4.6:_Produktcode–Generierung\|"Exercise 4.6"]] .

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	'''(3)'''  The basic structure of the product code is shown on the [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"Basic structure of a Product Code"]] ~~page~~.		'''(3)'''  The basic structure of the product code is shown in the [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"Basic structure of a Product Code"]] section.
	* You can see the information block with $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,		* 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 word length is the total number of all bits: $n = n_1 \cdot n_2 = 7 \cdot 6 \ \underline{= 42}$.

Hwang at 19:16, 1 November 2022

2022-11-01T19:16:29Z

Guenter at 16:52, 31 October 2022

2022-10-31T16:52:26Z

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	[[Category:Channel Coding: Exercises\|^4.2 ~~Grundlegendes zu den Produktcodes~~^]]		[[Category:Channel Coding: Exercises\|^4.2 About the Product Codes^]]

Guenter: Guenter moved page Aufgaben:Aufgabe 4.6Z: Grundlagen der Produktcodes to Aufgaben:Exercise 4.6Z: Basics of Product Codes

2022-10-31T16:47:58Z

Guenter moved page Aufgaben:Aufgabe 4.6Z: Grundlagen der Produktcodes to Aufgaben:Exercise 4.6Z: Basics of Product Codes

← Older revision		Revision as of 18:47, 31 October 2022
(No difference)

Noah at 15:05, 31 October 2022

2022-10-31T15:05:04Z

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	[[File:P_ID3002__KC_Z_4_6_v3.png\|right\|frame\|Generator matrices of the component codes]]		[[File:P_ID3002__KC_Z_4_6_v3.png\|right\|frame\|Generator matrices of the component codes]]
	We consider here a product code according to the description on page  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"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.		We consider here a product code according to the description on page  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"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.


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	'''(3)'''  The basic structure of the product code is shown on the [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"Basic structure of a ~~product code~~"]] page.		'''(3)'''  The basic structure of the product code is shown on the [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"Basic structure of a Product Code"]] page.
	* You can see the information block with $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,		* 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 codeword length is the total number of all bits: $n = n_1 \cdot n_2 = 7 \cdot 6 \ \underline{= 42}$.

← Older revision		Revision as of 18:19, 6 December 2022
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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]]		[[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  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"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.		We consider here a product code according to the description in section  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"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.






			<u>Hints:</u>
			*This exercise belongs to the chapter  [[Channel_Coding/The_Basics_of_Product_Codes\|"Basics of a Product Code"]].

			*Reference is made to the section  [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_Product_Code\|"Basic structure of a product code"]].

	~~Hints:~~		*The two component codes are also covered in the  [[Aufgaben:Aufgabe_4.6:_Produktcode–Generierung\|$\text{Exercise 4.6}$]] .
	*This exercise belongs to the chapter  [[Channel_Coding/The_Basics_of_Product_Codes\|"Basics of a Product Code"]].
	*Reference is made in particular to the section  [[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  [[Aufgaben:Aufgabe_4.6:_Produktcode–Generierung\|"Exercise 4.6"]] .

← Older revision		Revision as of 21:16, 1 November 2022
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	'''(1)'''  Correct are <u>statements 1, 2 and 4</u>:		'''(1)'''  Correct are <u>statements 1, 2 and 4</u>:
	* The number of rows of the generator matrix $\mathbf{G}_1$ indicates the length of the information block   ⇒   $k = 4$.		* 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 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 $4 &times 4$ diagonal matrix.		* The code is systematic because the generator matrix $\mathbf{G}_1$ starts with a $4 &times 4$ diagonal matrix.
	*This is a "normal" Hamming code.		*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.		*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 $\rm (7, \ 4, \ 3)$.		*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   ⇒   $d_{\rm min} = 3$.		*The last parameter in this code label specifies the minimum distance   ⇒   $d_{\rm min} = 3$.
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	'''(3)'''  The basic structure of the product code is shown on the [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"Basic structure of a Product Code"]] page.		'''(3)'''  The basic structure of the product code is shown on the [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code\|"Basic structure of a Product Code"]] page.
	* You can see the information block with $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,		* 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 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 $R = k/n = 12/42 = 2/7$.		*The code rate is thus given by $R = k/n = 12/42 = 2/7$.
	*Or:   $R = R_1 \cdot R_2 = 4/7 \cdot 1/2 \ \underline{= 2/7} \approx 0.289$.		*Or:   $R = R_1 \cdot R_2 = 4/7 \cdot 1/2 \ \underline{= 2/7} \approx 0.289$.