Aufgaben:Exercise 1.08Z: Equivalent Codes - Revision history

Hwang at 16:00, 23 January 2023

2023-01-23T16:00:29Z

Guenter at 11:51, 11 July 2022

2022-07-11T11:51:44Z

Guenter at 11:37, 11 July 2022

2022-07-11T11:37:08Z

Guenter at 15:53, 10 July 2022

2022-07-10T15:53:40Z

Noah at 13:07, 6 July 2022

2022-07-06T13:07:57Z

Noah at 12:57, 6 July 2022

2022-07-06T12:57:54Z

Guenter at 09:48, 11 May 2022

2022-05-11T09:48:47Z

Guenter: Guenter moved page Aufgaben:Aufgabe 1.08Z: Äquivalente Codes to Aufgaben:Exercise 1.08Z: Equivalent Codes

2022-05-11T09:40:28Z

Guenter moved page Aufgabe 1.08Z: Äquivalente Codes to Exercise 1.08Z: Equivalent Codes

Javier: Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

2021-03-23T12:49:05Z

Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

Javier: Text replacement - "[[Kanalcodierung" to "[[Channel_Coding"

2020-07-09T12:50:16Z

Text replacement - "[[Kanalcodierung" to "[[Channel_Coding"

@@ Line 32: / Line 32: @@
 *This exercise belongs to the chapter&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes|"General Description of Linear Block Codes"]].
-*Reference is made in particular to the pages&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Systematic_Codes|"Systematic Codes"]] &nbsp; and &nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Identical_Codes|"Identical Codes"]].
+*Reference is made in particular to the sections&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Systematic_Codes|"Systematic Codes"]] &nbsp; and &nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Identical_Codes|"Identical Codes"]].
 *Note that the specification of a parity-check matrix&nbsp; $\boldsymbol{\rm H}$&nbsp; is not unique.&nbsp; If one changes the order of the parity-check equations, this corresponds to a swapping of rows.

@@ Line 84: / Line 84: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are the <u>answers 1, 3 and 4</u>:
+'''(1)'''&nbsp; Correct are the&nbsp; <u>answers 1, 3 and 4</u>:
-*For a systematic (6, 3) block code, the following must hold:
+*For a systematic&nbsp; $(6, 3)$&nbsp; block code,&nbsp; the following must hold:
 :$$\underline{x} = ( x_1, x_2, x_3, x_4, x_5, x_6) = ( u_1, u_2, u_3, p_1, p_2, p_{3}) \hspace{0.05cm}.$$
-This condition is satisfied by code A, code C, and code D, but not by code B.
+*This condition is satisfied by code&nbsp; $\rm A$, code&nbsp; $\rm C$, and code&nbsp; $\rm D$, but not by code&nbsp; $\rm B$.
-'''(2)'''&nbsp; Correct is only <u>answer 1</u>:
+'''(2)'''&nbsp; Correct is only&nbsp; <u>answer 1</u>:
-*Only code A and code B are identical codes. They contain exactly the same code words and differ only by other assignments $\underline{u} \rightarrow \underline{x}$.
+*Only code&nbsp; $\rm A$&nbsp; and code&nbsp; $\rm B$&nbsp; are identical codes.&nbsp; They contain exactly the same code words and differ only by other assignments&nbsp; $\underline{u} \rightarrow \underline{x}$.
-*As indicated in the sample solution to [[Aufgaben:Exercise_1.08:_Identical_Codes|Exercise 1.8 (3)]], one gets from the generator matrix ${ \boldsymbol{\rm G}}_{\rm B}$ to the generator matrix ${ \boldsymbol{\rm G}}_{\rm A}$
-:*by swapping/permuting rows alone, or
+*As indicated in the solution to&nbsp; [[Aufgaben:Exercise_1.08:_Identical_Codes|"Exercise 1.8 (3)"]],&nbsp; one gets from the generator matrix&nbsp; ${ \boldsymbol{\rm G}}_{\rm B}$&nbsp; to the generator matrix&nbsp; ${ \boldsymbol{\rm G}}_{\rm A}$
 :*by replacing a row with the linear combination between that row and another.
-'''(3)'''&nbsp; Thus, the correct answer is <u>answer 2</u> alone:
+'''(3)'''&nbsp; Thus,&nbsp; the correct answer is&nbsp; <u>answer 2</u>&nbsp; alone:
-*Code A and code B are more than equivalent, namely identical.
+*Code&nbsp; $\rm A$&nbsp; and code&nbsp; $\rm B$&nbsp; are more than equivalent,&nbsp; namely identical.
-*Code C and D also differ, for example, by the minimum Hamming distance $d_{\rm min} = 3$ and $d_{\rm min} = 2$, respectively, and are thus also not equivalent.
-*Code B and code C, on the other hand, show the same properties, for example $d_{\rm min} = 3$ holds for both. However, they contain different codewords.
+*Code&nbsp; $\rm C$&nbsp; and code&nbsp; $\rm D$&nbsp; also differ,&nbsp; for example,&nbsp; by the minimum Hamming distance&nbsp; $d_{\rm min} = 3$&nbsp; and&nbsp; $d_{\rm min} = 2$,&nbsp; respectively,&nbsp; and are thus also not equivalent.
-*The last column of ${ \boldsymbol{\rm G}}_{\rm B}$ gives the first column of ${ \boldsymbol{\rm G}}_{\rm C}$.
+'''(4)'''&nbsp; Correct is&nbsp; <u>answer 3</u>:
 *The first column of ${ \boldsymbol{\rm G}}_{\rm B}$ gives the second column of ${ \boldsymbol{\rm G}}_{\rm C}$.
-*The second column of ${ \boldsymbol{\rm G}}_{\rm B}$ gives the third column of ${ \boldsymbol{\rm G}}_{\rm C}$, etc.
+*The last column of&nbsp; ${ \boldsymbol{\rm G}}_{\rm B}$&nbsp; gives the first column of&nbsp; ${ \boldsymbol{\rm G}}_{\rm C}$.
 '''(5)'''&nbsp; All statements are true</u>:
-*The condition ${ \boldsymbol{\rm H}} \cdot { \boldsymbol{\rm G}}^{\rm T} = \boldsymbol{0}$ holds for all linear codes.
+*The condition&nbsp; ${ \boldsymbol{\rm H}} \cdot { \boldsymbol{\rm G}}^{\rm T} = \boldsymbol{0}$&nbsp; holds for all linear codes.
 {{ML-Fuß}}

@@ Line 51: / Line 51: @@
 |type="[]"}
-+ Code &nbsp;$\rm A$&nbsp; and code &nbsp;$\rm B$,
++ Code &nbsp;$\rm A$&nbsp; and&nbsp; code &nbsp;$\rm B$,
-- Code &nbsp;$\rm B$&nbsp; and code &nbsp;$\rm C$,
+- Code &nbsp;$\rm B$&nbsp; and&nbsp; code &nbsp;$\rm C$,
-- Code &nbsp;$\rm C$&nbsp; and code &nbsp;$\rm D$.
+- Code &nbsp;$\rm C$&nbsp; and&nbsp; code &nbsp;$\rm D$.
 {Which of the given code pairs are equivalent but not identical?
 |type="[]"}
-- Code &nbsp;$\rm A$&nbsp; and code &nbsp;$\rm B$,
+- Code &nbsp;$\rm A$&nbsp; and&nbsp; code &nbsp;$\rm B$,
-+ Code &nbsp;$\rm B$&nbsp; and code &nbsp;$\rm C$,
++ Code &nbsp;$\rm B$&nbsp; and&nbsp; code &nbsp;$\rm C$,
-- Code &nbsp;$\rm C$&nbsp; and code &nbsp;$\rm D$.
+- Code &nbsp;$\rm C$&nbsp; and&nbsp; code &nbsp;$\rm D$.
-{How do the generator matrices&nbsp; $G_{\rm B}$&nbsp; and&nbsp; $G_{\rm C}$ differ?
+{How do the generator matrices&nbsp; $G_{\rm B}$&nbsp; and&nbsp; $G_{\rm C}$&nbsp; differ?
 |type="[]"}
 - By different linear combinations of different rows.
 - By cyclic shifting of rows by &nbsp;$1$&nbsp; down.
-+ By cyclic shifting of columns by &nbsp;$1$&nbsp; to the right.?
++ By cyclic shifting of columns by &nbsp;$1$&nbsp; to the right?

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID2394__KC_Z_1_8.png|right|frame|Four &nbsp;$(6, 3)$ block codes]]
+[[File:P_ID2394__KC_Z_1_8.png|right|frame|Four &nbsp;$(6, 3)$&nbsp; block codes]]
-In the graph, the mappings&nbsp; $\underline{u} \rightarrow \underline{x}$&nbsp; for different codes are given, each characterized below by the generator matrix&nbsp; $\boldsymbol{\rm G}$&nbsp; and the parity-check matrix&nbsp; $\boldsymbol{\rm H}$&nbsp; respectively:
+In the graph,&nbsp; the mappings&nbsp; $\underline{u} \rightarrow \underline{x}$&nbsp; for different codes are given,&nbsp; each characterized below by the generator matrix&nbsp; $\boldsymbol{\rm G}$&nbsp; and the parity-check matrix&nbsp; $\boldsymbol{\rm H}$,&nbsp; respectively:
 *${\boldsymbol{\rm Code \ A}}$:
@@ Line 28: / Line 28: @@
-Hints :
+*Reference is made in particular to the pages&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Systematic_Codes|"Systematic Codes"]] &nbsp; and &nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Identical_Codes|"Identical Codes"]].
-*This exercise belongs to the chapter&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes|General Description of Linear Block Codes]].
+*Note that the specification of a parity-check matrix&nbsp; $\boldsymbol{\rm H}$&nbsp; is not unique.&nbsp; If one changes the order of the parity-check equations, this corresponds to a swapping of rows.

@@ Line 83: / Line 83: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig sind die <u>Antworten  1, 3 und 4</u>:
+'''(1)'''&nbsp; Correct are the <u>answers 1, 3 and 4</u>:
-*Für einen systematischen (6, 3)–Blockcode muss gelten:
+*For a systematic (6, 3) block code, the following must hold:
 :$$\underline{x} = ( x_1, x_2, x_3, x_4, x_5, x_6) = ( u_1, u_2, u_3, p_1, p_2, p_{3}) \hspace{0.05cm}.$$
-Diese Bedingung erfüllen Code A, Code C und Code D, nicht aber Code B.
+This condition is satisfied by code A, code C, and code D, but not by code B.
-'''(2)'''&nbsp; Richtig ist nur <u>Antwort  1</u>:
+'''(2)'''&nbsp; Correct is only <u>answer 1</u>:
-*Nur Code A und Code B sind identische Codes. Sie beinhalten genau die gleichen Codeworte und unterscheiden sich nur durch andere Zuordnungen $\underline{u} \rightarrow \underline{x}$.
+*Only code A and code B are identical codes. They contain exactly the same code words and differ only by other assignments $\underline{u} \rightarrow \underline{x}$.
-*Wie in der Musterlösung zur [[Aufgaben:1.08_Identische_Codes|Aufgabe A1.8 (3)]] angegeben, gelangt man von der Generatormatrix ${ \boldsymbol{\rm G}}_{\rm B}$  zur Generatormatrix ${ \boldsymbol{\rm G}}_{\rm A}$
+*As indicated in the sample solution to [[Aufgaben:Exercise_1.08:_Identical_Codes|Exercise 1.8 (3)]], one gets from the generator matrix ${ \boldsymbol{\rm G}}_{\rm B}$ to the generator matrix ${ \boldsymbol{\rm G}}_{\rm A}$
-:*allein durch Vertauschen/Permutieren von Zeilen, oder
+:*by swapping/permuting rows alone, or
-:*durch Ersetzen einer Zeile durch die Linearkombination zwischen dieser Zeile und einer anderen.
+:*by replacing a row with the linear combination between that row and another.
-'''(3)'''&nbsp; Richtig ist somit allein <u>Antwort 2</u>:
+'''(3)'''&nbsp; Thus, the correct answer is <u>answer 2</u> alone:
-*Code A und Code B sind mehr als äquivalent, nämlich identisch.
+*Code A and code B are more than equivalent, namely identical.
-*Code C und D unterscheiden sich zum Beispiel auch durch die minimale Hamming–Distanz $d_{\rm min} = 3$ bzw. $d_{\rm min} = 2$ und sind somit auch nicht äquivalent.
+*Code C and D also differ, for example, by the minimum Hamming distance $d_{\rm min} = 3$ and $d_{\rm min} = 2$, respectively, and are thus also not equivalent.
-*Code B und Code C zeigen dagegen  gleiche Eigenschaften, beispielsweise gilt für beide $d_{\rm min} = 3$. Sie beinhalten aber andere Codeworte.
+*Code B and code C, on the other hand, show the same properties, for example $d_{\rm min} = 3$ holds for both. However, they contain different codewords.
-'''(4)'''&nbsp; Richtig ist <u>Antwort 3</u>:
+'''(4)'''&nbsp; Correct is <u>answer 3</u>:
-*Die letzte Spalte von ${ \boldsymbol{\rm G}}_{\rm B}$ ergibt die erste Spalte von ${ \boldsymbol{\rm G}}_{\rm C}$.
+*The last column of ${ \boldsymbol{\rm G}}_{\rm B}$ gives the first column of ${ \boldsymbol{\rm G}}_{\rm C}$.
-*Die erste Spalte von ${ \boldsymbol{\rm G}}_{\rm B}$ ergibt die zweite Spalte von ${ \boldsymbol{\rm G}}_{\rm C}$.
+*The first column of ${ \boldsymbol{\rm G}}_{\rm B}$ gives the second column of ${ \boldsymbol{\rm G}}_{\rm C}$.
-*Die zweite Spalte von ${ \boldsymbol{\rm G}}_{\rm B}$ ergibt die dritte Spalte von ${ \boldsymbol{\rm G}}_{\rm C}$, usw.
+*The second column of ${ \boldsymbol{\rm G}}_{\rm B}$ gives the third column of ${ \boldsymbol{\rm G}}_{\rm C}$, etc.
-'''(5)'''&nbsp; Alle Aussagen treffen zu</u>:
+'''(5)'''&nbsp; All statements are true</u>:
-*Die Bedingung ${ \boldsymbol{\rm H}} \cdot { \boldsymbol{\rm G}}^{\rm T} = \boldsymbol{0}$ gilt für alle linearen Codes.
+*The condition ${ \boldsymbol{\rm H}} \cdot { \boldsymbol{\rm G}}^{\rm T} = \boldsymbol{0}$ holds for all linear codes.
 {{ML-Fuß}}

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Kanalcodierung/Allgemeine Beschreibung linearer Blockcodes
+{{quiz-Header|Buchseite=Channel_Coding/General_Description_of_Linear_Block_Codes
 }}
-[[File:P_ID2394__KC_Z_1_8.png|right|frame|Vier &nbsp;$(6, 3)$–Blockcodes]]
+[[File:P_ID2394__KC_Z_1_8.png|right|frame|Four &nbsp;$(6, 3)$ block codes]]
-In der Grafik sind die Zuordnungen&nbsp; $\underline{u} \rightarrow \underline{x}$&nbsp; für verschiedene Codes angegeben, die im Folgenden jeweils durch die Generatormatrix&nbsp; $\boldsymbol{\rm G}$&nbsp; und die Prüfmatrix&nbsp; $\boldsymbol{\rm H}$&nbsp; charakterisiert werden:
+In the graph, the mappings&nbsp; $\underline{u} \rightarrow \underline{x}$&nbsp; for different codes are given, each characterized below by the generator matrix&nbsp; $\boldsymbol{\rm G}$&nbsp; and the parity-check matrix&nbsp; $\boldsymbol{\rm H}$&nbsp; respectively:
 *${\boldsymbol{\rm Code \ A}}$:
@@ Line 19: / Line 19: @@
 :$${ \boldsymbol{\rm G}}_{\rm D} = \begin{pmatrix} 1 &0 &0 &1 &0 &1\\ 0 &1 &0 &1 &0 &0\\ 0 &0 &1 &0 &1 &0 \end{pmatrix} \hspace{0.05cm},\hspace{0.5cm}{ \boldsymbol{\rm H}}_{\rm D} = \begin{pmatrix} 1 &1 &0 &1 &0 &0\\ 0 &0 &1 &0 &1 &0\\ 1 &0 &0 &0 &0 &1 \end{pmatrix} \hspace{0.05cm}.$$
-In dieser Aufgabe soll untersucht werden, welche dieser Codes bzw. Codepaare
+This task is to investigate which of these codes or code pairs are
-*systematisch sind,
+*are systematic,
-*identisch sind&nbsp; (das heißt: &nbsp; Verschiedene Codes haben gleiche Codeworte),
+*are identical&nbsp; (that is: &nbsp; Different codes have same code words),
-*äquivalent sind&nbsp; (das heißt: &nbsp; Verschiedene Codes haben gleiche Codeparameter).
+*are equivalent&nbsp; (that is: &nbsp; Different codes have same code parameters).
@@ Line 30: / Line 30: @@
-''Hinweise'' :
+Hints :
-*Die Aufgabe gehört zum Kapitel&nbsp; [[Channel_Coding/Allgemeine_Beschreibung_linearer_Blockcodes|Allgemeine Beschreibung linearer Blockcodes]].
+*This exercise belongs to the chapter&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes|General Description of Linear Block Codes]].
-*Bezug genommen wird insbesondere auf die Seiten&nbsp; [[Channel_Coding/Allgemeine_Beschreibung_linearer_Blockcodes#Systematische_Codes|Systematische Codes]]&nbsp; sowie&nbsp; [[Channel_Coding/Allgemeine_Beschreibung_linearer_Blockcodes#Identische_Codes|Identische Codes]].
+*Reference is made in particular to the pages&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Systematic_Codes|Systematic Codes]]&nbsp; and&nbsp; [[Channel_Coding/General_Description_of_Linear_Block_Codes#Identical_Codes|Identical Codes]].
-*Anzumerken ist, dass die Angabe einer Prüfmatrix&nbsp; $\boldsymbol{\rm H}$&nbsp; nicht eindeutig ist.
+*Note that the specification of a parity-check matrix&nbsp; $\boldsymbol{\rm H}$&nbsp; is not unique.
-*Verändert man die Reihenfolge der Prüfgleichungen, so entspricht dies einer Vertauschung von Zeilen.
+*If one changes the order of the parity-check equations, this corresponds to a swapping of rows.
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Welche der nachfolgend aufgeführten Codes sind systematisch?
+{Which of the codes listed below are systematic?
 |type="[]"}
 + Code &nbsp;$\rm A$,
@@ Line 49: / Line 49: @@
 + Code &nbsp;$\rm D$.
-{Welche der vorgegebenen Codepaare sind identisch?
+{Which of the given code pairs are identical?
 |type="[]"}
-+ Code &nbsp;$\rm A$&nbsp; und Code &nbsp;$\rm B$,
++ Code &nbsp;$\rm A$&nbsp; and code &nbsp;$\rm B$,
-- Code &nbsp;$\rm B$&nbsp; und Code &nbsp;$\rm C$,
+- Code &nbsp;$\rm B$&nbsp; and code &nbsp;$\rm C$,
-- Code &nbsp;$\rm C$&nbsp; und Code &nbsp;$\rm D$.
+- Code &nbsp;$\rm C$&nbsp; and code &nbsp;$\rm D$.
-{Welche der gegebenen Codepaare sind äquivalent, aber nicht identisch?
+{Which of the given code pairs are equivalent but not identical?
 |type="[]"}
-- Code &nbsp;$\rm A$&nbsp; und Code &nbsp;$\rm B$,
+- Code &nbsp;$\rm A$&nbsp; and code &nbsp;$\rm B$,
-+ Code &nbsp;$\rm B$&nbsp; und Code &nbsp;$\rm C$,
++ Code &nbsp;$\rm B$&nbsp; and code &nbsp;$\rm C$,
-- Code &nbsp;$\rm C$&nbsp; und Code &nbsp;$\rm D$.
+- Code &nbsp;$\rm C$&nbsp; and code &nbsp;$\rm D$.
-{Wie unterscheiden sich die Generatormatrizen&nbsp; $G_{\rm B}$&nbsp; und&nbsp; $G_{\rm C}$?
+{How do the generator matrices&nbsp; $G_{\rm B}$&nbsp; and&nbsp; $G_{\rm C}$ differ?
 |type="[]"}
-- Durch verschiedene Linearkombinationen verschiedener Zeilen.
+- By different linear combinations of different rows.
-- Durch zyklische Vertauschung der Zeilen um &nbsp;$1$&nbsp; nach unten.
+- By cyclic shifting of rows by &nbsp;$1$&nbsp; down.
-+  Durch zyklische Vertauschung der Spalten um &nbsp;$1$&nbsp; nach rechts.
++ By cyclic shifting of columns by &nbsp;$1$&nbsp; to the right.?
-{Bei welchen Codes gilt&nbsp; ${ \boldsymbol{\rm H}} \cdot { \boldsymbol{\rm G}}^{\rm T} = \boldsymbol{0}$?
+{For which codes applies&nbsp; ${ \boldsymbol{\rm H}} \cdot { \boldsymbol{\rm G}}^{\rm T} = \boldsymbol{0}$?
 |type="[]"}
 + Code &nbsp;$\rm A$,

← Older revision		Revision as of 09:48, 11 May 2022
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−	[[Category:Channel Coding: Exercises\|^1.4 ~~Beschreibung linearer Blockcodes~~	+	[[Category:Channel Coding: Exercises\|^1.4 Linear Block Code Description

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← Older revision	Revision as of 09:40, 11 May 2022
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← Older revision		Revision as of 12:49, 23 March 2021
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−	[[Category:~~Aufgaben zu Kanalcodierung~~\|^1.4 Beschreibung linearer Blockcodes	+	[[Category:Channel Coding: Exercises\|^1.4 Beschreibung linearer Blockcodes

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