Aufgaben:Exercise 1.07Z: Classification of Block Codes - Revision history

Hwang at 15:59, 23 January 2023

2023-01-23T15:59:27Z

Guenter at 14:51, 30 September 2022

2022-09-30T14:51:20Z

Guenter at 15:20, 11 July 2022

2022-07-11T15:20:36Z

Guenter at 13:30, 10 July 2022

2022-07-10T13:30:49Z

Guenter at 13:08, 10 July 2022

2022-07-10T13:08:24Z

Guenter at 13:07, 10 July 2022

2022-07-10T13:07:46Z

Noah at 21:18, 30 June 2022

2022-06-30T21:18:20Z

Noah at 18:28, 30 June 2022

2022-06-30T18:28:55Z

Noah at 18:23, 30 June 2022

2022-06-30T18:23:35Z

Noah at 18:21, 30 June 2022

2022-06-30T18:21:45Z

← Older revision		Revision as of 15:59, 23 January 2023
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	*This exercise belongs to the chapter  [[Channel_Coding/General_Description_of_Linear_Block_Codes\|"General description of linear block codes"]].		*This exercise belongs to the chapter  [[Channel_Coding/General_Description_of_Linear_Block_Codes\|"General description of linear block codes"]].

−	*Reference is also made to the ~~pages~~  [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes\|"Single parity–check codes"]]  and [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes\|"Repetition codes"]].	+	*Reference is also made to the sections  [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes\|"Single parity–check codes"]]  and [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes\|"Repetition codes"]].

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID2391__KC_Z_1_7_neu.png|right|frame|Block codes of length&nbsp; $n = 4$ '''Korrektur''' code ]]
+[[File:P_ID2391__KC_Z_1_7_neu.png|right|frame|Block codes of length&nbsp; $n = 4$ ]]
 We consider block codes of length&nbsp; $n = 4$:

@@ Line 28: / Line 28: @@
 The individual codes are explicitly indicated in the graphic.&nbsp; The questions for these tasks are about the terms
-*[[Channel_Coding/General_Description_of_Linear_Block_Codes#Linear_codes_and_cyclic_codes|linear codes]],
+*[[Channel_Coding/General_Description_of_Linear_Block_Codes#Linear_codes_and_cyclic_codes|"linear codes"]],
-*[[Channel_Coding/General_Description_of_Linear_Block_Codes#Systematic_Codes|systematic codes]],
+*[[Channel_Coding/General_Description_of_Linear_Block_Codes#Systematic_Codes|"systematic codes"]],
-*[[Channel_Coding/General_Description_of_Linear_Block_Codes#Representation_of_SPC_and_RC_as_dual_codes|dual codes]].
+*[[Channel_Coding/General_Description_of_Linear_Block_Codes#Representation_of_SPC_and_RC_as_dual_codes|"dual codes"]].

@@ Line 81: / Line 81: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; <u>Statements 1 and 2</u> are correct:
+'''(1)'''&nbsp; <u>Statements 1 and 2</u>&nbsp; are correct:
-* That is why there are $\rm 4 \ over \ 2 = 6$ codewords.
+* That is why there are&nbsp; "$\rm 4 \ over \ 2 = 6$"&nbsp; code words.
-* Statement 3 is false. For example, if the first bit is $0$, there is one codeword starting $00$ and two codewords starting $01$.
+* Statement 3 is false.&nbsp; For example,&nbsp; if the first bit is&nbsp; "$0$",&nbsp; there is one code word starting&nbsp; "$00$"&nbsp; and two code words starting&nbsp; "$01$".
-'''(2)'''&nbsp; <u>Statements 1 to 4</u> are correct:
+'''(2)'''&nbsp; <u>Statements 1 to 4</u>&nbsp; are correct:
-* All codes that can be described by a generator matrix $\boldsymbol {\rm G}$ and/or a parity-check matrix $\boldsymbol {\rm H}$ are linear.
+* All codes that can be described by a generator matrix&nbsp; $\boldsymbol {\rm G}$&nbsp; and/or a parity-check matrix&nbsp; $\boldsymbol {\rm H}$&nbsp; are linear.
-*In contrast, "Code 5" does not satisfy any of the conditions required for linear codes. For example
+*In contrast,&nbsp; "code 5" does not satisfy any of the conditions required for linear codes.&nbsp; For example
-:*is missing the all zero word,
+:*is missing the all-zero word,
-:*the code cardinality $|\mathcal{C}|$ is not a power of two,
+:*the code size&nbsp; $|\mathcal{C}|$&nbsp; is not a power of two,
-:*gives $(0, 1, 0, 1) \oplus (1, 0, 1, 0) = (1, 1, 1, 1)$ no valid codeword.
+:*gives&nbsp; $(0, 1, 0, 1) \oplus (1, 0, 1, 0) = (1, 1, 1, 1)$&nbsp; no valid code word.
-'''(3)'''&nbsp; <u>Statements 1 to 3</u> are correct:
+'''(3)'''&nbsp; <u>Statements 1 to 3</u>&nbsp; are correct:
-*In a systematic code, the first $k$ bits of each codeword $\underline{x}$ must always be equal to the information word $\underline{u}$.
+*In a systematic code,&nbsp; the first&nbsp; $k$&nbsp; bits of each code word&nbsp; $\underline{x}$&nbsp; must always be equal to the information word&nbsp; $\underline{u}$.
-*This is achieved if the beginning of the generator matrix $\boldsymbol {\rm G}$ is a unit matrix $\boldsymbol{\rm I}_{k}$.
+*This is achieved if the beginning of the generator matrix&nbsp; $\boldsymbol {\rm G}$&nbsp; is an identity matrix&nbsp; $\boldsymbol{\rm I}_{k}$.
-*This is true for "Code 1" (with dimension $k = 3$), "Code 2" (with $k = 1$) and "code 3" (with $k = 2$).
+*This is true for&nbsp; "code 1"&nbsp; $($with dimension&nbsp; $k = 3)$,&nbsp; "code 2" $($with&nbsp; $k = 1)$&nbsp; and&nbsp; "code 3"&nbsp; $($with $k = 2)$.
-*The generator matrix of "Code 2", however, is not explicitly stated. It is:
+*The generator matrix of&nbsp; "code 2",&nbsp; however,&nbsp; is not explicitly stated.&nbsp; It is:
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$
-'''(4)'''&nbsp; <u>Statement 1</u> is correct:
+'''(4)'''&nbsp; <u>Statement 1</u>&nbsp; is correct:
-*Dual codes are those where the parity-check matrix $\boldsymbol {\rm H}$ of one code is equal to the generator matrix $\boldsymbol {\rm G}$ of the other code.
+*Dual codes are those where the parity-check matrix&nbsp; $\boldsymbol {\rm H}$&nbsp; of one code is equal to the generator matrix&nbsp; $\boldsymbol {\rm G}$&nbsp; of the other code.
-*For example, this is true for "Code 1" and "Code 2."
+*For example,&nbsp; this is true for "code 1" and "code 2."
-*For the SPC (4, 3) holds:
+*For the&nbsp; $\text{SPC (4, 3)}$&nbsp; holds:
 :$${ \boldsymbol{\rm H}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}, \hspace{0.3cm} { \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$
-:and for the repetition code RC (4, 1):
+:and for the repetition code&nbsp; $\text{RC (4, 1)}$:
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}, \hspace{0.3cm} { \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$
-*Statement 2 is certainly wrong, already for dimensional reasons: The generator matrix $\boldsymbol {\rm G}$ of "Code 3" is a $2×4$ matrix and the parity-check matrix $\boldsymbol {\rm H}$ of "Code 2" is a $3×4$ matrix.
+*Statement 2 is certainly wrong,&nbsp; already for dimensional reasons:&nbsp; The&nbsp; $\boldsymbol {\rm G}$&nbsp; matrix  of&nbsp; "code 3"&nbsp; is a&nbsp; $2×4$&nbsp; matrix and the&nbsp; $\boldsymbol {\rm H}$&nbsp; matrix of&nbsp; "code 2"&nbsp; is a $3×4$&nbsp; matrix.
-*"Code 3" and "Code 4" also do not satisfy the conditions of dual codes. The parity-check equations of
+*"Code 3"&nbsp; and&nbsp; "code 4"&nbsp; also do not satisfy the conditions of dual codes.&nbsp; The parity-check equations of
 :$${\rm Code}\hspace{0.15cm}3 = \{ (0, 0, 0, 0) \hspace{0.05cm},\hspace{0.1cm} (0, 1, 1, 0) \hspace{0.05cm},\hspace{0.1cm}(1, 0, 0, 1) \hspace{0.05cm},\hspace{0.1cm}(1, 1, 1, 1) \}$$
@@ Line 127: / Line 127: @@
 :$$x_1 \oplus x_4 = 0\hspace{0.05cm},\hspace{0.2cm}x_2 \oplus x_3 = 0 \hspace{0.3cm} \Rightarrow\hspace{0.3cm} { \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &1 &0 \end{pmatrix} \hspace{0.05cm}.$$
-:In contrast, the generator matrix of "Code 4" is given as follows:
+:In contrast,&nbsp; the generator matrix of&nbsp; "code 4"&nbsp; is given as follows:
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &0 &0\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$

← Older revision		Revision as of 13:08, 10 July 2022
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	*This exercise belongs to the chapter  [[Channel_Coding/General_Description_of_Linear_Block_Codes\|"General description of linear block codes"]].		*This exercise belongs to the chapter  [[Channel_Coding/General_Description_of_Linear_Block_Codes\|"General description of linear block codes"]].

−	*Reference is also made to the pages  [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes\|"~~single~~ parity–check codes"]]  and [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes\|"~~repetition~~ codes"]].	+	*Reference is also made to the pages  [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes\|"Single parity–check codes"]]  and [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes\|"Repetition codes"]].

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID2391__KC_Z_1_7_neu.png|right|frame|Block codes of length&nbsp; $n = 4$ ]]
+[[File:P_ID2391__KC_Z_1_7_neu.png|right|frame|Block codes of length&nbsp; $n = 4$ '''Korrektur''' code ]]
 We consider block codes of length&nbsp; $n = 4$:
-*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|single parity–check]]&nbsp; code&nbsp; $\text{SPC (4, 3)}$ &nbsp; &rArr; &nbsp; "Code 1" &nbsp; with the generator matrix
+*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|single parity–check code]]&nbsp;  $\text{SPC (4, 3)}$ &nbsp; &rArr; &nbsp; "code 1" &nbsp; with the generator matrix
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$
-*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|repetition code]] &nbsp; $\text{RC (4, 1)}$ &nbsp; &rArr; &nbsp; "Code 2" &nbsp; with the parity-check matrix
+*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|repetition code]]&nbsp; $\text{RC (4, 1)}$ &nbsp; &rArr; &nbsp; "code 2" &nbsp; with the parity-check matrix
 :$${ \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$
-*the &nbsp; $\text{(4, 2)}$ block code &nbsp; &rArr; &nbsp; "Code 3" &nbsp; with the generator matrix
+*the&nbsp; $\text{(4, 2)}$&nbsp; block code &nbsp; &rArr; &nbsp; "code 3" &nbsp; with the generator matrix
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &1 &1 \end{pmatrix} \hspace{0.05cm},$$
-*the&nbsp; $\text{(4, 2)}$ block code &nbsp; &rArr; &nbsp; "Code 4" &nbsp; with the generator matrix
+*the&nbsp; $\text{(4, 2)}$&nbsp; block code &nbsp; &rArr; &nbsp; "code 4" &nbsp; with the generator matrix
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &0 &0\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$
-*another "Code 5" &nbsp; with the code cardinality (Überstetzung von Codeumfang laut Wachter-Zeh VL)&nbsp; $|\hspace{0.05cm}C\hspace{0.05cm}| = 6$.
+*another "code 5" &nbsp; with code size&nbsp; $|\hspace{0.05cm}C\hspace{0.05cm}| = 6$.
-The individual codes are explicitly indicated in the graphic. The questions for these tasks are about the terms
+The individual codes are explicitly indicated in the graphic.&nbsp; The questions for these tasks are about the terms
 *[[Channel_Coding/General_Description_of_Linear_Block_Codes#Linear_codes_and_cyclic_codes|linear codes]],
@@ Line 37: / Line 37: @@
+*Reference is also made to the pages&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|"single parity&ndash;check codes"]]&nbsp; and [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|"repetition codes"]].
@@ Line 51: / Line 49: @@
 <quiz display=simple>
-{How can "Code 5" be described?
+{How can&nbsp; "code 5"&nbsp; be described?
 |type="[]"}
-+ There are exactly two zeros in each codeword.
++ There are exactly two zeros in each code word.
-+ There are exactly two ones in each codeword.
++ There are exactly two ones in each code word.
-- After each $0$, the symbols&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; are equally likely.
+- After each&nbsp; "$0$",&nbsp; the symbols&nbsp; "$0$"&nbsp; and&nbsp; "$1$"&nbsp; are equally likely.
 {Which of the following block codes are linear?

@@ Line 51: / Line 51: @@
 <quiz display=simple>
-{Wie lässt sich "Code 5" beschreiben?
+{How can "Code 5" be described?
 |type="[]"}
-+ In jedem Codewort sind genau zwei Nullen enthalten.
++ There are exactly two zeros in each codeword.
-+ In jedem Codewort sind genau zwei Einsen enthalten.
++ There are exactly two ones in each codeword.
-- Nach jeder $0$ sind die Symbole&nbsp;  $0$&nbsp;  und&nbsp;  $1$&nbsp;  gleichwahrscheinlich.
+- After each $0$, the symbols&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; are equally likely.
-{Welche der folgenden Blockcodes sind linear?
+{Which of the following block codes are linear?
 |type="[]"}
-+ Code 1,
++ code 1,
-+ Code 2,
++ code 2,
-+ Code 3,
++ code 3,
-+ Code 4,
++ code 4,
-- Code 5.
+- code 5.
-{Welche der folgenden Blockcodes sind systematisch?
+{Which of the following block codes are systematic?
 |type="[]"}
-+ Code 1,
++ code 1,
-+ Code 2,
++ code 2,
-+ Code 3,
++ code 3,
-- Code 4,
+- code 4,
-- Code 5.
+- code 5.
-{Welche Codepaare sind zueinander dual?
+{Which code pairs are dual to each other?
 |type="[]"}
-+ Code 1 und Code 2,
++ code 1 and code 2,
-- Code 2 und Code 3,
+- code 2 and code 3,
-- Code 3 und Code 4.
+- code 3 and code 4.
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig sind die <u>Aussagen 1 und 2</u>:
+'''(1)'''&nbsp; <u>Statements 1 and 2</u> are correct:
-* Deshalb gibt es auch $\rm 4 \ über \ 2 = 6$ Codeworte.
+* That is why there are $\rm 4 \ over \ 2 = 6$ codewords.
-* Aussage 3 ist falsch. Ist zum Beispiel das erste Bit $0$, so gibt es ein Codewort mit dem Beginn $00$ und zwei Codeworte, die mit $01$ beginnen.
+* Statement 3 is false. For example, if the first bit is $0$, there is one codeword starting $00$ and two codewords starting $01$.
-'''(2)'''&nbsp; Richtig sind die <u>Aussagen 1 bis 4</u>:
+'''(2)'''&nbsp; <u>Statements 1 to 4</u> are correct:
-* Alle Codes, die durch eine Generatormatrix $\boldsymbol {\rm G}$ und/oder eine Prüfmatrix $\boldsymbol {\rm H}$ beschrieben werden können, sind linear.
+* All codes that can be described by a generator matrix $\boldsymbol {\rm G}$ and/or a parity-check matrix $\boldsymbol {\rm H}$ are linear.
-*Dagegen erfüllt "Code 5" keine der für lineare Codes erforderlichen Bedingungen. Beispielsweise
+*In contrast, "Code 5" does not satisfy any of the conditions required for linear codes. For example
-:*fehlt das Nullwort,
+:*is missing the all zero word,
-:*ist der Codeumfang $|\mathcal{C}|$ keine Zweierpotenz,
+:*the code cardinality $|\mathcal{C}|$ is not a power of two,
-:*ergibt $(0, 1, 0, 1) \oplus (1, 0, 1, 0) = (1, 1, 1, 1)$ kein gültiges Codewort.
+:*gives $(0, 1, 0, 1) \oplus (1, 0, 1, 0) = (1, 1, 1, 1)$ no valid codeword.
-'''(3)'''&nbsp; Richtig sind die <u>Aussagen 1 bis 3</u>:
+'''(3)'''&nbsp; <u>Statements 1 to 3</u> are correct:
-*Bei einem systematischen Code müssen stets die ersten $k$ Bit eines jeden Codewortes $\underline{x}$ gleich dem Informationswort $\underline{u}$ sein.
+*In a systematic code, the first $k$ bits of each codeword $\underline{x}$ must always be equal to the information word $\underline{u}$.
-*Dies wird erreicht, wenn der Beginn der Generatormatrix $\boldsymbol {\rm G}$ eine Einheitsmatrix $\boldsymbol{\rm I}_{k}$ darstellt.
+*This is achieved if the beginning of the generator matrix $\boldsymbol {\rm G}$ is a unit matrix $\boldsymbol{\rm I}_{k}$.
-*Dies trifft für "Code 1" (mit Dimension $k = 3$), "Code 2" (mit $k = 1$) und "Code 3" (mit $k = 2$) zu.
+*This is true for "Code 1" (with dimension $k = 3$), "Code 2" (with $k = 1$) and "code 3" (with $k = 2$).
-*Die Generatormatrix von "Code 2" ist allerdings nicht explizit angegeben. Sie lautet:
+*The generator matrix of "Code 2", however, is not explicitly stated. It is:
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$
-'''(4)'''&nbsp; Richtig ist die <u>Aussage 1</u>:
+'''(4)'''&nbsp; <u>Statement 1</u> is correct:
-*Von dualen Codes spricht man, wenn die Prüfmatrix $\boldsymbol {\rm H}$ des einen Codes gleich der Generatormatrix $\boldsymbol {\rm G}$ des anderen Codes ist.
+*Dual codes are those where the parity-check matrix $\boldsymbol {\rm H}$ of one code is equal to the generator matrix $\boldsymbol {\rm G}$ of the other code.
-*Dies trifft zum Beispiel für "Code 1" und "Code 2" zu.
+*For example, this is true for "Code 1" and "Code 2."
-*Für den SPC (4, 3) gilt:
+*For the SPC (4, 3) holds:
 :$${ \boldsymbol{\rm H}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}, \hspace{0.3cm} { \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$
-:und für den Wiederholungscode RC (4, 1):
+:and for the repetition code RC (4, 1):
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}, \hspace{0.3cm} { \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$
-*Aussage 2 ist mit Sicherheit falsch, schon aus Dimensionsgründen: Die Generatormatrix $\boldsymbol {\rm G}$ von "Code 3" ist eine $2×4$–Matrix und die Prüfmatrix $\boldsymbol {\rm H}$ von "Code 2" eine $3×4$–Matrix.
+*Statement 2 is certainly wrong, already for dimensional reasons: The generator matrix $\boldsymbol {\rm G}$ of "Code 3" is a $2×4$ matrix and the parity-check matrix $\boldsymbol {\rm H}$ of "Code 2" is a $3×4$ matrix.
-*"Code 3" und "Code 4" erfüllen ebenfalls nicht die Bedingungen dualer Codes. Die Prüfgleichungen von
+*"Code 3" and "Code 4" also do not satisfy the conditions of dual codes. The parity-check equations of
 :$${\rm Code}\hspace{0.15cm}3 = \{ (0, 0, 0, 0) \hspace{0.05cm},\hspace{0.1cm} (0, 1, 1, 0) \hspace{0.05cm},\hspace{0.1cm}(1, 0, 0, 1) \hspace{0.05cm},\hspace{0.1cm}(1, 1, 1, 1) \}$$
-:lauten:
+:are as follows:
 :$$x_1 \oplus x_4 = 0\hspace{0.05cm},\hspace{0.2cm}x_2 \oplus x_3 = 0 \hspace{0.3cm} \Rightarrow\hspace{0.3cm} { \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &1 &0 \end{pmatrix} \hspace{0.05cm}.$$
-:Dagegen ist die Generatormatrix von "Code 4" wie folgt gegeben:
+:In contrast, the generator matrix of "Code 4" is given as follows:
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &0 &0\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$

← Older revision		Revision as of 18:28, 30 June 2022
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	*This exercise belongs to the chapter  [[Channel_Coding/General_Description_of_Linear_Block_Codes\|General description of linear block codes]].		*This exercise belongs to the chapter  [[Channel_Coding/General_Description_of_Linear_Block_Codes\|General description of linear block codes]].
−	*Reference is also made to the pages  [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes\|single parity–check codes]]  and[[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes\|repetition codes]].	+	*Reference is also made to the pages  [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes\|single parity–check codes]]  and [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes\|repetition codes]].

@@ 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_ID2391__KC_Z_1_7_neu.png|right|frame|Blockcodes der Länge&nbsp; $n = 4$ ]]
+[[File:P_ID2391__KC_Z_1_7_neu.png|right|frame|Block codes of length&nbsp; $n = 4$ ]]
-Wir betrachten Blockcodes der Länge&nbsp; $n = 4$:
+We consider block codes of length&nbsp; $n = 4$:
-*den&nbsp; [[Channel_Coding/Beispiele_binärer_Blockcodes#Single_Parity.E2.80.93check_Codes|Single Parity–check]]&nbsp; Code&nbsp; $\text{SPC (4, 3)}$ &nbsp; &rArr; &nbsp; "Code 1" &nbsp; mit der Generatormatrix
+*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|single parity–check]]&nbsp; code&nbsp; $\text{SPC (4, 3)}$ &nbsp; &rArr; &nbsp; "code 1" &nbsp; with the generator matrix
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$
-*den&nbsp; [[Channel_Coding/Beispiele_binärer_Blockcodes#Wiederholungscodes|Wiederholungscode]] &nbsp; $\text{RC (4, 1)}$ &nbsp; &rArr; &nbsp; "Code 2" &nbsp; mit der Prüfmatrix
+*the&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|repetition code]] &nbsp; $\text{RC (4, 1)}$ &nbsp; &rArr; &nbsp; "code 2" &nbsp; with the parity-check matrix
 :$${ \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$
-*den&nbsp; $\text{(4, 2)}$–Blockcode &nbsp; &rArr; &nbsp; "Code 3" &nbsp; mit der Generatormatrix
+*the &nbsp; $\text{(4, 2)}$ block code &nbsp; &rArr; &nbsp; "code 3" &nbsp; with the generator matrix
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &1 &1 \end{pmatrix} \hspace{0.05cm},$$
-*den&nbsp; $\text{(4, 2)}$–Blockcode &nbsp; &rArr; &nbsp; "Code 4" &nbsp; mit der Generatormatrix
+*the&nbsp; $\text{(4, 2)}$ block code &nbsp; &rArr; &nbsp; "code 4" &nbsp; with the generator matrix
 :$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &0 &0\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$
-*einen weiteren "Code 5" &nbsp; mit dem Codeumfang&nbsp; $|\hspace{0.05cm}C\hspace{0.05cm}| = 6$.
+*another "Code 5" &nbsp; with the code cardinality (Überstetzung von Codeumfang laut Wachter-Zeh VL)&nbsp; $|\hspace{0.05cm}C\hspace{0.05cm}| = 6$.
-In der Grafik sind die einzelnen Codes explizit angegegeben. Bei den Fragen zu diesen Aufgaben geht es um die Begriffe
+The individual codes are explicitly indicated in the graphic. The questions for these tasks are about the terms
-*[[Channel_Coding/Allgemeine_Beschreibung_linearer_Blockcodes#Lineare_Codes_und_zyklische_Codes|lineare Codes]],
+*[[Channel_Coding/General_Description_of_Linear_Block_Codes#Linear_codes_and_cyclic_codes|linear codes]],
-*[[Channel_Coding/Allgemeine_Beschreibung_linearer_Blockcodes#Systematische_Codes|systematische Codes]],
+*[[Channel_Coding/General_Description_of_Linear_Block_Codes#Systematic_Codes|systematic codes]],
-*[[Channel_Coding/Allgemeine_Beschreibung_linearer_Blockcodes#Darstellung_von_SPC_und_RC_als_duale_Codes|duale Codes]].
+*[[Channel_Coding/General_Description_of_Linear_Block_Codes#Representation_of_SPC_and_RC_as_dual_codes|dual codes]].
@@ Line 40: / Line 40: @@
-''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 aber auchauf die Seiten&nbsp; [[Channel_Coding/Beispiele_binärer_Blockcodes#Single_Parity.E2.80.93check_Codes|Single Parity&ndash;check Codes]]&nbsp; sowie [[Channel_Coding/Beispiele_binärer_Blockcodes#Wiederholungscodes|Wiederholungscodes]].
+*Reference is also made to the pages&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|single parity&ndash;check codes]]&nbsp; and[[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|repetition codes]].
-===Fragebogen===
+===Questions===
 <quiz display=simple>