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

Guenter at 16:29, 6 December 2022

2022-12-06T16:29:29Z

Guenter at 16:21, 6 December 2022

2022-12-06T16:21:12Z

Guenter at 16:19, 6 December 2022

2022-12-06T16:19:58Z

Hwang at 23:41, 12 November 2022

2022-11-12T23:41:38Z

Hwang at 19:16, 1 November 2022

2022-11-01T19:16:29Z

Guenter at 16:52, 31 October 2022

2022-10-31T16:52:26Z

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 Aufgabe 4.6Z: Grundlagen der Produktcodes to Exercise 4.6Z: Basics of Product Codes

Noah at 15:05, 31 October 2022

2022-10-31T15:05:04Z

Noah at 15:04, 31 October 2022

2022-10-31T15:04:32Z

Noah at 14:58, 31 October 2022

2022-10-31T14:58:21Z

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 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are <u>statements 1, 2 and 4</u>:
+'''(1)'''&nbsp; Correct are the&nbsp; <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 &nbsp; &#8658; &nbsp; $k = 4$.
+* The number of rows of the generator matrix&nbsp; $\mathbf{G}_1$&nbsp; indicates the length of the information block &nbsp; &#8658; &nbsp; $k = 4$.
 * The code word length is equal to the number of columns &nbsp; &#8658; &nbsp; $n=4$ &nbsp; &#8658; &nbsp; 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&nbsp; $\mathbf{G}_1$&nbsp; starts with a&nbsp; $4 &times 4$&nbsp; diagonal matrix.
 *For this, with the code word length $n$ and the number of check bits &nbsp; &#8658; &nbsp; $m = n - k$, the relation $n = 2^m - 1$ holds.
-*In the present case, this is the (normal) Hamming code $\rm (7, \ 4, \ 3)$.
+*This is a&nbsp; "normal"&nbsp; Hamming code.
 *The last parameter in this code label specifies the minimum distance &nbsp; &#8658; &nbsp; $d_{\rm min} = 3$.
-'''(2)'''&nbsp; Correct <u>statements 2, 3 and 4</u>:
+'''(2)'''&nbsp; Correct are the&nbsp; <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&nbsp; $n = 6, \ k = 3$&nbsp; and&nbsp; $d_{\rm min} = 3$,&nbsp; also in systematic form.
 *The code rate is $R = 1/2$.
-'''(3)'''&nbsp; 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)'''&nbsp; The basic structure of the product code is shown in the&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code|"Basic structure of a Product Code"]]&nbsp; 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&nbsp; $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,
 * 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: &nbsp; $R = R_1 \cdot R_2 = 4/7 \cdot 1/2 \ \underline{= 2/7} \approx 0.289$.
 * The free distance is $d = d_1 \cdot d_2 = 3 \cdot 3 \ \underline{= 9}$.
 {{ML-Fuß}}

← Older revision		Revision as of 16:21, 6 December 2022
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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\|$\text{Exercise 4.6}$]] .

@@ Line 1: / Line 1: @@
 {{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&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.
+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.
-Hints:
+*The two component codes are also covered in the&nbsp; [[Aufgaben:Aufgabe_4.6:_Produktcode–Generierung|$\text{Exercise 4.6}$]]&nbsp;.

@@ Line 2: / Line 2: @@
 [[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&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.
+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.
@@ Line 13: / Line 13: @@
 Hints:
 *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 page&nbsp; [[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&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;.
@@ Line 61: / Line 61: @@
-'''(3)'''&nbsp; 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)'''&nbsp; 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 $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,
 * The code word length is the total number of all bits: $n = n_1 \cdot n_2 = 7 \cdot 6 \ \underline{= 42}$.

@@ Line 48: / Line 48: @@
 '''(1)'''&nbsp; 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 &nbsp; &#8658; &nbsp; $k = 4$.
-* The codeword length is equal to the number of columns &nbsp; &#8658; &nbsp; $n=4$ &nbsp; &#8658; &nbsp; Code rate $R = k/n = 4/7$.
+* The code word length is equal to the number of columns &nbsp; &#8658; &nbsp; $n=4$ &nbsp; &#8658; &nbsp; 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.
-*For this, with the codeword length $n$ and the number of check bits &nbsp; &#8658; &nbsp; $m = n - k$, the relation $n = 2^m - 1$ holds.
+*For this, with the code word length $n$ and the number of check bits &nbsp; &#8658; &nbsp; $m = n - k$, 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 &nbsp; &#8658; &nbsp; $d_{\rm min} = 3$.
@@ Line 63: / Line 63: @@
 '''(3)'''&nbsp; 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 $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,
-* 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 $R = k/n = 12/42 = 2/7$.
 *Or: &nbsp; $R = R_1 \cdot R_2 = 4/7 \cdot 1/2 \ \underline{= 2/7} \approx 0.289$.

← Older revision		Revision as of 16:52, 31 October 2022
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−	[[Category:Channel Coding: Exercises\|^4.2 ~~Grundlegendes zu den Produktcodes~~^]]	+	[[Category:Channel Coding: Exercises\|^4.2 About the Product Codes^]]

← Older revision	Revision as of 16:47, 31 October 2022
(No difference)

@@ Line 12: / Line 12: @@
 Hints:
-*The exercise belongs to the chapter&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes|"Basics of a Product Code"]].
+*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 page&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;.
@@ Line 22: / Line 22: @@
 ===Questions===
 <quiz display=simple>
-{Welche Aussagen erlaubt die Generatormatrix&nbsp; $\mathbf{G}_1$&nbsp; über den Code&nbsp; $\mathcal{C}_1$?
+{What statements does the generator matrix&nbsp; $\mathbf{G}_1$&nbsp; allow about the code&nbsp; $\mathcal{C}_1$?
 |type="[]"}
-+ Die Coderate von&nbsp; $\mathcal{C}_1$&nbsp; ist&nbsp; $R_1 = 4/7$.
++ The code rate of&nbsp; $\mathcal{C}_1$&nbsp; is&nbsp; $R_1 = 4/7$.
-+ Der Code&nbsp; $\mathcal{C}_1$&nbsp; ist systematisch.
++ The code&nbsp; $\mathcal{C}_1$&nbsp; is systematic.
-- $\mathcal{C}_1$&nbsp; ist ein verkürzter Hamming&ndash;Code.
+- $\mathcal{C}_1$&nbsp; is a truncated Hamming code.
-+ Die minimale Distanz dieses Codes ist&nbsp; $d_1 = 3$.
++ The minimum distance of this code is&nbsp; $d_1 = 3$.
-{Welche Aussagen erlaubt die Generatormatrix&nbsp; $\mathbf{G}_2$&nbsp; über den Code&nbsp; $\mathcal{C}_2$?
+{What statements does the generator matrix&nbsp; $\mathbf{G}_2$&nbsp; allow about the code&nbsp; $\mathcal{C}_2$?
 |type="[]"}
-- Die Coderate von&nbsp; $\mathcal{C}_2$&nbsp; ist&nbsp; $R_2 = 4/7$.
+- The code rate of&nbsp; $\mathcal{C}_2$&nbsp; is&nbsp; $R_2 = 4/7$.
-+ Der Code&nbsp; $\mathcal{C}_2$&nbsp; ist systematisch.
++ The code&nbsp; $\mathcal{C}_2$&nbsp; is systematic.
-+ $\mathcal{C}_2$&nbsp; ist ein verkürzter Hamming&ndash;Code.
++ $\mathcal{C}_2$&nbsp; is a truncated Hamming code.
-+ Die minimale Distanz dieses Codes ist&nbsp; $d_2 = 3$.
++ The minimum distance of this code is&nbsp; $d_2 = 3$.
-{Geben Sie die Parameter des Produktcodes&nbsp; $\mathcal{C} = \mathcal{C}_1 &times \mathcal{C}_2$&nbsp; an.
+{Specify the parameters of the product code&nbsp; $\mathcal{C} = \mathcal{C}_1 &times \mathcal{C}_2$&nbsp;.
 |type="{}"}
 $k \hspace{0.25cm} = \ ${ 12 3% }
@@ Line 44: / Line 44: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig sind die <u>Aussagen 1, 2 und 4</u>:
+'''(1)'''&nbsp; Correct are <u>statements 1, 2 and 4</u>:
-* Die Anzahl der Zeilen der Generatormatrix $\mathbf{G}_1$ gibt die Länge des Informationsblocks an &nbsp; &#8658; &nbsp;  $k = 4$.
+* The number of rows of the generator matrix $\mathbf{G}_1$ indicates the length of the information block &nbsp; &#8658; &nbsp; $k = 4$.
-* Die Codewortlänge  ist gleich der Anzahl der Spalten &nbsp; &#8658; &nbsp; $n=4$ &nbsp; &#8658; &nbsp; Coderate $R = k/n = 4/7$.
+* The codeword length is equal to the number of columns &nbsp; &#8658; &nbsp; $n=4$ &nbsp; &#8658; &nbsp; Code rate $R = k/n = 4/7$.
-* Der Code ist systematisch, da die Generatormatrix $\mathbf{G}_1$ mit einer $4 &times 4$&ndash;Diagonalmatrix beginnt.
+* The code is systematic because the generator matrix $\mathbf{G}_1$ starts with a $4 &times 4$ diagonal matrix.
-* Es handelt sich um einen "normalen" Hammingcode.
+*This is a "normal" Hamming code.
-*Für diesen gilt mit der Codewortlänge $n$ und der Anzahl der Prüfbits &nbsp; &#8658; &nbsp; $m = n - k$ der Zusammenhang $n = 2^m - 1$.
+*For this, with the codeword length $n$ and the number of check bits &nbsp; &#8658; &nbsp; $m = n - k$, the relation $n = 2^m - 1$ holds.
-* Im vorliegenden Fall handelt es sich um den (normalen) Hammingcode $\rm (7, \ 4, \ 3)$.
+*In the present case, this is the (normal) Hamming code $\rm (7, \ 4, \ 3)$.
-*Der letzte Parameter in dieser Codebezeichnung gibt die minimale Distanz an &nbsp; &#8658; &nbsp; $d_{\rm min} = 3$.
+*The last parameter in this code label specifies the minimum distance &nbsp; &#8658; &nbsp; $d_{\rm min} = 3$.
-'''(2)'''&nbsp; Richtig sind die <u>Aussagen 2, 3 und 4</u>:
+'''(2)'''&nbsp; Correct <u>statements 2, 3 and 4</u>:
-*Es handelt sich um einen verkürzten Hammingcode mit dem Parameter $n = 6, \ k = 3$ und $d_{\rm min} = 3$, ebenfalls in systematischer Form.
+*This is a truncated Hamming code with parameter $n = 6, \ k = 3$ and $d_{\rm min} = 3$, also in systematic form.
-*Die Coderate beträgt $R = 1/2$.
+*The code rate is $R = 1/2$.
-'''(3)'''&nbsp; Die Grundstruktur des Produktcodes ist auf der Seite [[Channel_Coding/Grundlegendes_zu_den_Produktcodes#Grundstruktur_eines_Produktcodes|Grundstruktur eines Produktcodes]] dargestellt.
+'''(3)'''&nbsp; 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.
-* Man erkennt den Informationsblock mit $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,
+* You can see the information block with $k = k_1 \cdot k_2 = 4 \cdot 3 \ \underline{= 12}$,
-* Die Codewortlänge ist die Gesamtzahl aller Bit: $n = n_1 \cdot n_2 = 7 \cdot 6 \ \underline{= 42}$.
+* The codeword length is the total number of all bits: $n = n_1 \cdot n_2 = 7 \cdot 6 \ \underline{= 42}$.
-* Die Coderate ergibt sich somit zu $R = k/n = 12/42 = 2/7$.
+*The code rate is thus given by $R = k/n = 12/42 = 2/7$.
-*Oder: &nbsp; $R = R_1 \cdot R_2 = 4/7 \cdot 1/2 \ \underline{= 2/7} \approx 0.289$.
+*Or: &nbsp; $R = R_1 \cdot R_2 = 4/7 \cdot 1/2 \ \underline{= 2/7} \approx 0.289$.
-* Die freie Distanz beträgt $d = d_1 \cdot d_2 = 3 \cdot 3 \ \underline{= 9}$.
+* The free distance is $d = d_1 \cdot d_2 = 3 \cdot 3 \ \underline{= 9}$.
 {{ML-Fuß}}