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

Maintenance script: Fix interlanguage link: resolve redirect chain

2026-03-16T15:55:34Z

Fix interlanguage link: resolve redirect chain

← Older revision		Revision as of 17:55, 16 March 2026
Line 139:		Line 139:

	^]]		^]]
	[[de:Aufgaben:1.~~07Z_Klassifizierung_von_Blockcodes~~]]		[[de:Aufgaben:Aufgabe 1.07Z: Klassifizierung von Blockcodes]]

Maintenance script: Add German interlanguage link

2026-03-16T12:29:56Z

Add German interlanguage link

← Older revision		Revision as of 14:29, 16 March 2026
Line 139:		Line 139:

	^]]		^]]
			[[de:Aufgaben:1.07Z_Klassifizierung_von_Blockcodes]]

Hwang at 15:59, 23 January 2023

2023-01-23T15:59:27Z

← Older revision		Revision as of 17:59, 23 January 2023
Line 41:		Line 41:
	*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"]].

Guenter at 14:51, 30 September 2022

2022-09-30T14:51:20Z

← Older revision		Revision as of 16:51, 30 September 2022
Line 3:		Line 3:
	}}		}}

	[[File:P_ID2391__KC_Z_1_7_neu.png\|right\|frame\|Block codes of length  $n = 4$ ~~'''Korrektur''' code~~ ]]		[[File:P_ID2391__KC_Z_1_7_neu.png\|right\|frame\|Block codes of length  $n = 4$ ]]

	We consider block codes of length  $n = 4$:		We consider block codes of length  $n = 4$:

Guenter at 15:20, 11 July 2022

2022-07-11T15:20:36Z

← Older revision		Revision as of 17:20, 11 July 2022
Line 28:		Line 28:
	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.  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"]].

Guenter at 13:30, 10 July 2022

2022-07-10T13:30:49Z

← Older revision		Revision as of 15:30, 10 July 2022
Line 81:		Line 81:
	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  <u>Statements 1 and 2</u> are correct:		'''(1)'''  <u>Statements 1 and 2</u>  are correct:
	* That is why there are $\rm 4 \ over \ 2 = 6$ ~~codewords~~.		* That is why there are  "$\rm 4 \ over \ 2 = 6$"  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.  For example,  if the first bit is  "$0$",  there is one code word starting  "$00$"  and two code words starting  "$01$".



	'''(2)'''  <u>Statements 1 to 4</u> are correct:		'''(2)'''  <u>Statements 1 to 4</u>  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  $\boldsymbol {\rm G}$  and/or a parity-check matrix  $\boldsymbol {\rm H}$  are linear.
	*In contrast, "~~Code~~ 5" does not satisfy any of the conditions required for linear codes. For example		*In contrast,  "code 5" does not satisfy any of the conditions required for linear codes.  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  $\|\mathcal{C}\|$  is not a power of two,
	:*gives $(0, 1, 0, 1) \oplus (1, 0, 1, 0) = (1, 1, 1, 1)$ no valid ~~codeword~~.		:*gives  $(0, 1, 0, 1) \oplus (1, 0, 1, 0) = (1, 1, 1, 1)$  no valid code word.



	'''(3)'''  <u>Statements 1 to 3</u> are correct:		'''(3)'''  <u>Statements 1 to 3</u>  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,  the first  $k$  bits of each code word  $\underline{x}$  must always be equal to the information word  $\underline{u}$.
	*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  $\boldsymbol {\rm G}$  is an identity matrix  $\boldsymbol{\rm I}_{k}$.
	*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  "code 1"  $($with dimension  $k = 3)$,  "code 2" $($with  $k = 1)$  and  "code 3"  $($with $k = 2)$.
	*The generator matrix of "~~Code~~ 2", however, is not explicitly stated. It is:		*The generator matrix of  "code 2",  however,  is not explicitly stated.  It is:
	:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$		:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$



	'''(4)'''  <u>Statement 1</u> is correct:		'''(4)'''  <u>Statement 1</u>  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  $\boldsymbol {\rm H}$  of one code is equal to the generator matrix  $\boldsymbol {\rm G}$  of the other code.
	*For example, this is true for "~~Code~~ 1" and "~~Code~~ 2."		*For example,  this is true for "code 1" and "code 2."
	*For the SPC (4, 3) holds:		*For the  $\text{SPC (4, 3)}$  holds:

	:$${ \boldsymbol{\rm H}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}, \hspace{0.3cm} { \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$		:$${ \boldsymbol{\rm H}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}, \hspace{0.3cm} { \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$

	:and for the repetition code RC (4, 1):		:and for the repetition code  $\text{RC (4, 1)}$:

	:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}, \hspace{0.3cm} { \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$		:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &1 &1 \end{pmatrix} \hspace{0.05cm}, \hspace{0.3cm} { \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$

	*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,  already for dimensional reasons:  The  $\boldsymbol {\rm G}$  matrix of  "code 3"  is a  $2×4$  matrix and the  $\boldsymbol {\rm H}$  matrix of  "code 2"  is a $3×4$  matrix.

	*"Code 3" and "~~Code~~ 4" also do not satisfy the conditions of dual codes. The parity-check equations of		*"Code 3"  and  "code 4"  also do not satisfy the conditions of dual codes.  The parity-check equations of

	:$${\rm Code}\hspace{0.15cm}3 = \{ (0, 0, 0, 0) \hspace{0.05cm},\hspace{0.1cm} (0, 1, 1, 0) \hspace{0.05cm},\hspace{0.1cm}(1, 0, 0, 1) \hspace{0.05cm},\hspace{0.1cm}(1, 1, 1, 1) \}$$		:$${\rm Code}\hspace{0.15cm}3 = \{ (0, 0, 0, 0) \hspace{0.05cm},\hspace{0.1cm} (0, 1, 1, 0) \hspace{0.05cm},\hspace{0.1cm}(1, 0, 0, 1) \hspace{0.05cm},\hspace{0.1cm}(1, 1, 1, 1) \}$$
Line 127:		Line 127:
	:$$x_1 \oplus x_4 = 0\hspace{0.05cm},\hspace{0.2cm}x_2 \oplus x_3 = 0 \hspace{0.3cm} \Rightarrow\hspace{0.3cm} { \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &1 &0 \end{pmatrix} \hspace{0.05cm}.$$		:$$x_1 \oplus x_4 = 0\hspace{0.05cm},\hspace{0.2cm}x_2 \oplus x_3 = 0 \hspace{0.3cm} \Rightarrow\hspace{0.3cm} { \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &1 &0 \end{pmatrix} \hspace{0.05cm}.$$

	:In contrast, the generator matrix of "~~Code~~ 4" is given as follows:		:In contrast,  the generator matrix of  "code 4"  is given as follows:

	:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &0 &0\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$		:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &0 &0\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm}.$$

Guenter at 13:08, 10 July 2022

2022-07-10T13:08:24Z

← Older revision		Revision as of 15:08, 10 July 2022
Line 41:		Line 41:
	*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"]].

Guenter at 13:07, 10 July 2022

2022-07-10T13:07:46Z

← Older revision		Revision as of 15:07, 10 July 2022
Line 3:		Line 3:
	}}		}}

	[[File:P_ID2391__KC_Z_1_7_neu.png\|right\|frame\|Block codes of length  $n = 4$ ]]		[[File:P_ID2391__KC_Z_1_7_neu.png\|right\|frame\|Block codes of length  $n = 4$ '''Korrektur''' code ]]

	We consider block codes of length  $n = 4$:		We consider block codes of length  $n = 4$:

	*the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes\|single parity–check]]  ~~code ~~ $\text{SPC (4, 3)}$   ⇒   "~~Code~~ 1"   with the generator matrix		*the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes\|single parity–check code]]  $\text{SPC (4, 3)}$   ⇒   "code 1"   with the generator matrix

	:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$		:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$

	*the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes\|repetition code]]   $\text{RC (4, 1)}$   ⇒   "~~Code~~ 2"   with the parity-check matrix		*the  [[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes\|repetition code]]  $\text{RC (4, 1)}$   ⇒   "code 2"   with the parity-check matrix

	:$${ \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$		:$${ \boldsymbol{\rm H}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &0 &1\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$

	*the   $\text{(4, 2)}$ block code   ⇒   "~~Code~~ 3"   with the generator matrix		*the  $\text{(4, 2)}$  block code   ⇒   "code 3"   with the generator matrix

	:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &1 &1 \end{pmatrix} \hspace{0.05cm},$$		:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &0 &0 &1\\ 0 &1 &1 &1 \end{pmatrix} \hspace{0.05cm},$$

	*the  $\text{(4, 2)}$ block code   ⇒   "~~Code~~ 4"   with the generator matrix		*the  $\text{(4, 2)}$  block code   ⇒   "code 4"   with the generator matrix

	:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &0 &0\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$		:$${ \boldsymbol{\rm G}} = \begin{pmatrix} 1 &1 &0 &0\\ 0 &0 &1 &1 \end{pmatrix} \hspace{0.05cm},$$

	*another "~~Code~~ 5"   with ~~the~~ code ~~cardinality (Überstetzung von Codeumfang laut Wachter-Zeh VL)~~  $\|\hspace{0.05cm}C\hspace{0.05cm}\| = 6$.		*another "code 5"   with code size  $\|\hspace{0.05cm}C\hspace{0.05cm}\| = 6$.


	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.  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]],
Line 37:		Line 37:


			Hints :

			*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"]].
	~~Hints :~~

	*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]].


Line 51:		Line 49:

	<quiz display=simple>		<quiz display=simple>
	{How can "~~Code~~ 5" be described?		{How can  "code 5"  be described?
	\|type="[]"}		\|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  $0$  and  $1$  are equally likely.		- After each  "$0$",  the symbols  "$0$"  and  "$1$"  are equally likely.

	{Which of the following block codes are linear?		{Which of the following block codes are linear?

Noah at 21:18, 30 June 2022

2022-06-30T21:18:20Z

Show changes

Noah at 18:28, 30 June 2022

2022-06-30T18:28:55Z

← Older revision		Revision as of 20:28, 30 June 2022
Line 43:		Line 43:

	*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]].