Aufgaben:Exercise 4.6: Product Code Generation - Revision history

Guenter at 16:09, 6 December 2022

2022-12-06T16:09:13Z

Guenter at 15:49, 6 December 2022

2022-12-06T15:49:57Z

Guenter at 16:50, 30 November 2022

2022-11-30T16:50:13Z

Hwang at 12:46, 17 November 2022

2022-11-17T12:46:29Z

Hwang at 19:16, 1 November 2022

2022-11-01T19:16:00Z

Noah at 18:07, 31 October 2022

2022-10-31T18:07:50Z

Noah at 18:07, 31 October 2022

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Noah at 17:00, 31 October 2022

2022-10-31T17:00:23Z

Guenter at 16:51, 31 October 2022

2022-10-31T16:51:45Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.6: Generierung von Produktcodes to Aufgaben:Exercise 4.6: Product Code Generation

2022-10-31T16:45:04Z

Guenter moved page Aufgabe 4.6: Generierung von Produktcodes to Exercise 4.6: Product Code Generation

@@ Line 46: / Line 46: @@
+<u>Hints:</u>
 *This exercise belongs to the chapter&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes|"Basics of Product Code"]].
@@ Line 84: / Line 81: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are the <u>proposed solutions 1 and 3</u>:
+'''(1)'''&nbsp; Correct are the&nbsp; <u>proposed solutions 1 and 3</u>:&nbsp; In general:&nbsp;
+:$$\underline{x} = \underline{u} \cdot \mathbf{G}.$$
-In general $\underline{x} = \underline{u} \cdot \mathbf{G}$. From this follows for
+From this follows for
-* the first row vector:
+:* the first row vector:
 :$$\begin{pmatrix}
 &1 &1 &0
@@ Line 101: / Line 98: @@
 	\end{pmatrix} \hspace{0.05cm},$$
-* the second row vector:
+:* the second row vector:
 :$$\begin{pmatrix}
 &0 &0 &0
@@ Line 115: / Line 112: @@
 	\end{pmatrix} \hspace{0.05cm},$$
-* the third row vector:
+:* the third row vector:
 :$$\begin{pmatrix}
 &1 &1 &0
@@ Line 130: / Line 127: @@
-'''(2)'''&nbsp; Correct are the <u>proposed solutions 1, 2 and 4</u>:
+'''(2)'''&nbsp; Correct are the&nbsp; <u>proposed solutions 1, 2 and 4</u>:
 :$$\begin{pmatrix}
 &0 &1
@@ Line 155: / Line 152: @@
 To this subtask is to be noted further:
-* The given first column is correct if only because it coincides with a row (the third) of the generator matrix $\mathbf{G}_2$.
+# The given first column is correct if only because it coincides with a row&nbsp; $($the third$)$&nbsp; of the generator matrix $\mathbf{G}_2$.
-* The third column of the 2D code word should be identical to the second column, since the same code word $(1, \, 0, \, 1)$ is assumed.
+# The third column of the two-dimensional code word should be identical to the second column,&nbsp; since the same code word&nbsp; $(1, \, 0, \, 1)$&nbsp; is assumed.
-* However, the given vector $(1, \, 1, \, 0, \, 0, \, 1, \, 1)$ cannot be correct if only because $\mathcal{C}_2$ is a systematic code just like $\mathcal{C}_1$.
+# However,&nbsp; the given vector&nbsp; $(1, \, 1, \, 0, \, 0, \, 1, \, 1)$&nbsp; cannot be correct if only because&nbsp; $\mathcal{C}_2$&nbsp; is a systematic code just like&nbsp; $\mathcal{C}_1$.
-* Also the truncated $(6, \ 3, \ 3)$&ndash;Hamming code $C_2$ is linear, so that the assignment $\underline{u} = (0, \, 0, \, 0) \ \Rightarrow \ \ \underline{x} = (0, \, 0, \, 0, \, 0)$ can be stated without calculation.
+# Also the truncated&nbsp; $(6, \ 3, \ 3)$&nbsp; Hamming code $C_2$&nbsp; is linear,&nbsp; so that the assignment&nbsp; $\underline{u} = (0, \, 0, \, 0) \ \Rightarrow \ \ \underline{x} = (0, \, 0, \, 0, \, 0)$&nbsp; can be stated without calculation.
   [[File:EN_KC_A_4_6_c.png|right|frame|Complete code tables]]
-'''(3)'''&nbsp; The complete code tables
+'''(3)'''&nbsp; Given on the right are the complete code tables
-* of the Hamming code $(7, \ 4, \ 3)$, and
+* of the Hamming code $(7, \ 4, \ 3)$,&nbsp; and
-* of the truncated Hamming code $(6, \ 3, \ 3)$ are given on the right.
 One can see from this (without it being of interest for this exercise) that the codes considered here each have Hamming distance $d_{\rm min} = 3$.
 [[File:P_ID3012__KC_A_4_6d_v3.png|left|frame|Wanted product code]]
-<br><br>The left graph shows the result of the whole coding. At the bottom right you can see the checks&ndash;on&ndash;checks matrix of dimension $3 &times 3$.
+<br><br>
 <br clear=all>
-Concerning the subtask (3) the <u>suggested solutions 1 and 2</u> are correct:
+Concerning the subtask&nbsp; '''(3)'''&nbsp; the&nbsp; <u>suggested solutions 1 and 2</u>&nbsp; are correct:
 *It is a coincidence that here in the checks&ndash;on&ndash;checks matrix two rows and two columns are identical.
 *It doesn't matter whether rows 4 to 6 of the total matrix are obtained using the code $\mathcal{C}_1$ or columns 5 to 7 are obtained using the code $\mathcal{C}_2$.
 {{ML-Fuß}}

@@ Line 2: / Line 2: @@
 [[File:EN_KC_A_4_6.png|right|frame|Used component codes]]
-A&nbsp; $\rm product\:code \ (42, \ 12)$&nbsp; shall be generated, based on the following component codes:
+A&nbsp; $\rm product\:code \ (42, \ 12)$&nbsp; shall be generated,&nbsp; based on the following component codes:
 * the Hamming code&nbsp; $\rm HC \ (7, \ 4, \ 3)$&nbsp; &rArr; &nbsp; $\mathcal{C}_1$,
 * the truncated Hamming code $\rm HC \ (6, \ 3, \ 3)$ &nbsp; &rArr; &nbsp; $\mathcal{C}_2$.
-The code word belonging to an information block&nbsp; $\underline{u}$&nbsp; generally results according to the equation&nbsp; $\underline{x} = \underline{u} \cdot \mathbf{G}$. As in the&nbsp; [[Aufgaben:Aufgabe_4.6Z:_Grundlagen_der_Produktcodes|"Exercise 4.6Z"]]&nbsp; following generator matrices are assumed here:
+Corresponding code tables are given on the right,&nbsp; with three rows incomplete in each case.&nbsp; These are to be completed by you.
 :$${ \boldsymbol{\rm G}}_1
 =  \begin{pmatrix}
@@ Line 23: / Line 27: @@
 	\end{pmatrix} \hspace{0.05cm}.$$
-Throughout the exercise, apply to the information block:
+Throughout the exercise,&nbsp; apply to the information block:
 :$${ \boldsymbol{\rm U}}
 =  \begin{pmatrix}
@@ Line 32: / Line 36: @@
 Searched for according to the nomenclature in section&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code|"Basic structure of a product code"]]:
-* the parity-check matrix&nbsp; $\mathbf{P}^{(1)}$&nbsp; with respect to the horizontal code&nbsp; $\mathcal{C}_1$,
+* the parity-check matrix &nbsp; $\mathbf{P}^{(1)}$ &nbsp; with respect to the horizontal code&nbsp; $\mathcal{C}_1$,
 * the parity-check matrix&nbsp; $\mathbf{P}^{(2)}$&nbsp; with respect to the vertical code&nbsp; $\mathcal{C}_2$,
 * the checks&ndash;on&ndash;checks matrix&nbsp; $\mathbf{P}^{(12)}$.
@@ Line 44: / Line 50: @@
 Hints:
-*This exercise belongs to the chapter&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes|"Basic structure of a product code"]].
+*This exercise belongs to the chapter&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes|"Basics of 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_4.6Z:_Grundlagen_der_Produktcodes|"Exercise 4.6Z"]]&nbsp;.
+*Reference is also  made to the section&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code|"Basic structure of a product code"]].
@@ Line 62: / Line 70: @@
 {What are the results of column coding with the&nbsp; $(6, \ 3, \ 3)$ code&nbsp; $\mathcal{C}_2$?
 |type="[]"}
-+ 1. row: &nbsp; $\underline{u} = (0, \, 0, \, 1) \ \Rightarrow \ \underline{x} = (0, \, 0, \, 1, \, 0, \, 1, \, 1)$.
++ 1. column: &nbsp; $\underline{u} = (0, \, 0, \, 1) \ \Rightarrow \ \underline{x} = (0, \, 0, \, 1, \, 0, \, 1, \, 1)$.
-+ 2. row: &nbsp; $\underline{u} = (1, \, 0, \, 1) \ \Rightarrow \ \underline{x} = (1, \, 0, \, 1, \, 1, \, 0, \, 1)$.
++ 2. column: &nbsp; $\underline{u} = (1, \, 0, \, 1) \ \Rightarrow \ \underline{x} = (1, \, 0, \, 1, \, 1, \, 0, \, 1)$.
-- 3. row: &nbsp; $\underline{u} = (1, \, 0, \, 1) \ \Rightarrow \ \underline{x} = (1, \, 1, \, 0, \, 0, \, 1, \, 1)$.
+- 3. column: &nbsp; $\underline{u} = (1, \, 0, \, 1) \ \Rightarrow \ \underline{x} = (1, \, 1, \, 0, \, 0, \, 1, \, 1)$.
-+ 4. row: &nbsp; $\underline{u} = (0, \, 0, \, 0) \ \Rightarrow \ \underline{x} = (0, \, 0, \, 0, \, 0, \,0, \, 0)$.
++ 4. column: &nbsp; $\underline{u} = (0, \, 0, \, 0) \ \Rightarrow \ \underline{x} = (0, \, 0, \, 0, \, 0, \,0, \, 0)$.
 {What statements apply to the checks&ndash;on&ndash;checks matrix?
 |type="[]"}
-+ The first row is&nbsp; $(1, \, 0, \, 1)$&nbsp; and the first column&nbsp; $(1, \, 1, \, 0)$.
++ The first row is&nbsp; $(1, \, 0, \, 1)$&nbsp; and the first column is&nbsp; $(1, \, 1, \, 0)$.
-+ The second row is&nbsp; $(1, \, 0, \, 1)$&nbsp; and the second column&nbsp; $(0, \, 0, \, 0)$.
++ The second row is&nbsp; $(1, \, 0, \, 1)$&nbsp; and the second column is&nbsp; $(0, \, 0, \, 0)$.
-- The third row is&nbsp; $(0, \, 0, \, 0)$&nbsp; and the third column&nbsp; $(0, \, 0, \, 0)$.
+- The third row is&nbsp; $(0, \, 0, \, 0)$&nbsp; and the third column is&nbsp; $(0, \, 0, \, 0)$.
 </quiz>

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/The_Basics_of_Product_Codes}}
-[[File:P_ID3003__KC_A_4_6_v2.png|right|frame|Used component codes]]
+[[File:EN_KC_A_4_6.png|right|frame|Used component codes]]
 A&nbsp; $\rm product\:code \ (42, \ 12)$&nbsp; shall be generated, based on the following component codes:
 * the Hamming code&nbsp; $\rm HC \ (7, \ 4, \ 3)$&nbsp; &rArr; &nbsp; $\mathcal{C}_1$,
@@ Line 153: / Line 153: @@
-  [[File:P_ID3005__KC_A_4_6c_v1.png|right|frame|Complete code tables]]
+  [[File:EN_KC_A_4_6_c.png|right|frame|Complete code tables]]
 '''(3)'''&nbsp; The complete code tables

@@ Line 31: / Line 31: @@
 	\end{pmatrix} \hspace{0.05cm}.$$
-Searched for according to the nomenclature on page&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code|"Basic structure of a product code"]]:
+Searched for according to the nomenclature in section&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes#Basic_structure_of_a_product_code|"Basic structure of a product code"]]:
 * the parity-check matrix&nbsp; $\mathbf{P}^{(1)}$&nbsp; with respect to the horizontal code&nbsp; $\mathcal{C}_1$,
 * the parity-check matrix&nbsp; $\mathbf{P}^{(2)}$&nbsp; with respect to the vertical code&nbsp; $\mathcal{C}_2$,
@@ Line 45: / Line 45: @@
 Hints:
 *This exercise belongs to the chapter&nbsp; [[Channel_Coding/The_Basics_of_Product_Codes|"Basic structure 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_4.6Z:_Grundlagen_der_Produktcodes|"Exercise 4.6Z"]]&nbsp;.

@@ Line 8: / Line 8: @@
 The corresponding code tables are given on the right, with three rows incomplete in each case. These are to be completed by you.
-The codeword belonging to an information block&nbsp; $\underline{u}$&nbsp; generally results according to the equation&nbsp; $\underline{x} = \underline{u} \cdot \mathbf{G}$. As in the&nbsp; [[Aufgaben:Aufgabe_4.6Z:_Grundlagen_der_Produktcodes|"Exercise 4.6Z"]]&nbsp; following generator matrices are assumed here:
+The code word belonging to an information block&nbsp; $\underline{u}$&nbsp; generally results according to the equation&nbsp; $\underline{x} = \underline{u} \cdot \mathbf{G}$. As in the&nbsp; [[Aufgaben:Aufgabe_4.6Z:_Grundlagen_der_Produktcodes|"Exercise 4.6Z"]]&nbsp; following generator matrices are assumed here:
 :$${ \boldsymbol{\rm G}}_1
 =  \begin{pmatrix}
@@ Line 148: / Line 148: @@
 To this subtask is to be noted further:
 * The given first column is correct if only because it coincides with a row (the third) of the generator matrix $\mathbf{G}_2$.
-* The third column of the 2D codeword should be identical to the second column, since the same codeword $(1, \, 0, \, 1)$ is assumed.
+* The third column of the 2D code word should be identical to the second column, since the same code word $(1, \, 0, \, 1)$ is assumed.
 * However, the given vector $(1, \, 1, \, 0, \, 0, \, 1, \, 1)$ cannot be correct if only because $\mathcal{C}_2$ is a systematic code just like $\mathcal{C}_1$.
 * Also the truncated $(6, \ 3, \ 3)$&ndash;Hamming code $C_2$ is linear, so that the assignment $\underline{u} = (0, \, 0, \, 0) \ \Rightarrow \ \ \underline{x} = (0, \, 0, \, 0, \, 0)$ can be stated without calculation.

← Older revision		Revision as of 16:51, 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 16:45, 31 October 2022
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