Aufgaben:Exercise 1.11: Syndrome Decoding - Revision history

Maintenance script: Fix interlanguage link: resolve redirect chain

2026-03-16T15:58:23Z

Fix interlanguage link: resolve redirect chain

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	[[Category:Channel Coding: Exercises\|^1.5 Linear Block Code Decoding^]]		[[Category:Channel Coding: Exercises\|^1.5 Linear Block Code Decoding^]]
	[[de:Aufgaben:1.~~11_Syndromdecodierung~~]]		[[de:Aufgaben:Aufgabe 1.11: Syndromdecodierung]]

2026-03-16T12:30:02Z

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	[[Category:Channel Coding: Exercises\|^1.5 Linear Block Code Decoding^]]		[[Category:Channel Coding: Exercises\|^1.5 Linear Block Code Decoding^]]
			[[de:Aufgaben:1.11_Syndromdecodierung]]

2023-04-21T12:03:26Z

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	[[File:EN_KC_A_1_11d_v2.png\|right\|frame\|All coset leaders of the  $HC (7, \, 4, \, 3)$ ]]		[[File:EN_KC_A_1_11d_v2.png\|right\|frame\|All coset leaders of the  $\rm HC (7, \, 4, \, 3)$ ]]
	'''(5)'''  A comparison with the solution to the last subtask shows that  $(0, 1, 0, 0, 0, 0, 0) \cdot \boldsymbol{\rm H}^{\rm T}$  as a syndrome gives the second row of the transposed matrix:		'''(5)'''  A comparison with the solution to the last subtask shows that  $(0, 1, 0, 0, 0, 0, 0) \cdot \boldsymbol{\rm H}^{\rm T}$  as a syndrome gives the second row of the transposed matrix:

2023-04-20T16:07:49Z

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	[[File:~~P_ID2398__KC_A_1_11d~~.png\|right\|frame\|All coset leaders of the  $HC (7, \, 4, \, 3)$ ~~'''Korrektur'''~~ ]]		[[File:EN_KC_A_1_11d_v2.png\|right\|frame\|All coset leaders of the  $HC (7, \, 4, \, 3)$ ]]
	'''(5)'''  A comparison with the solution to the last subtask shows that  $(0, 1, 0, 0, 0, 0, 0) \cdot \boldsymbol{\rm H}^{\rm T}$  as a syndrome gives the second row of the transposed matrix:		'''(5)'''  A comparison with the solution to the last subtask shows that  $(0, 1, 0, 0, 0, 0, 0) \cdot \boldsymbol{\rm H}^{\rm T}$  as a syndrome gives the second row of the transposed matrix:

2023-03-13T16:09:33Z

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	[[File:P_ID2398__KC_A_1_11d.png\|right\|frame\|All coset leaders of the  $(7, \, 4, \, 3)$~~  Hamming code~~ '''Korrektur''' ]]		[[File:P_ID2398__KC_A_1_11d.png\|right\|frame\|All coset leaders of the  $HC (7, \, 4, \, 3)$ '''Korrektur''' ]]
	'''(5)'''  A comparison with the solution to the last subtask shows that  $(0, 1, 0, 0, 0, 0, 0) \cdot \boldsymbol{\rm H}^{\rm T}$  as a syndrome gives the second row of the transposed matrix:		'''(5)'''  A comparison with the solution to the last subtask shows that  $(0, 1, 0, 0, 0, 0, 0) \cdot \boldsymbol{\rm H}^{\rm T}$  as a syndrome gives the second row of the transposed matrix:

2023-01-23T15:32:02Z

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	'''(6)'''  For the  $(7, \, 4, \, 3)$  Hamming code under consideration,  the correct information word is decided if at most one bit within the code word is ~~corrupted~~ during transmission.  It follows:		'''(6)'''  For the  $(7, \, 4, \, 3)$  Hamming code under consideration,  the correct information word is decided if at most one bit within the code word is falsified during transmission.  It follows:

	:$${ \rm Pr(block\:errors)} \hspace{-0.15cm}\ = \ \hspace{-0.15cm} { \rm Pr(\ge 2\hspace{0.15cm}bit\hspace{0.15cm}errors\hspace{0.15cm})} =1 - { \rm Pr(0\hspace{0.15cm}bit\hspace{0.15cm}errors)} - { \rm Pr(1\hspace{0.15cm}bit\hspace{0.15cm}errors)} =1 - 0.9^7 - 7 \cdot 0.1 \cdot 0.9^7 \hspace{0.15cm} \underline{ = 0.15} \hspace{0.05cm}.$$		:$${ \rm Pr(block\:errors)} \hspace{-0.15cm}\ = \ \hspace{-0.15cm} { \rm Pr(\ge 2\hspace{0.15cm}bit\hspace{0.15cm}errors\hspace{0.15cm})} =1 - { \rm Pr(0\hspace{0.15cm}bit\hspace{0.15cm}errors)} - { \rm Pr(1\hspace{0.15cm}bit\hspace{0.15cm}errors)} =1 - 0.9^7 - 7 \cdot 0.1 \cdot 0.9^7 \hspace{0.15cm} \underline{ = 0.15} \hspace{0.05cm}.$$

2022-11-01T16:58:04Z

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	:$$\underline{s} = \underline{y} \cdot { \boldsymbol{\rm H}}^{\rm T} \in {\rm GF}(2^m) \hspace{0.05cm}.$$		:$$\underline{s} = \underline{y} \cdot { \boldsymbol{\rm H}}^{\rm T} \in {\rm GF}(2^m) \hspace{0.05cm}.$$

	*In  [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC\|"BSC channel"]]  the received word   $\underline{y} = \underline{x} \, ({\rm ~~codeword~~}) + \underline{e} \, ({\rm error\:vector})$   is also an element of  ${\rm GF}(2^n)$,  and it holds because of   $ \underline{x} · \boldsymbol {{\rm H} }^{\rm T} =		*In  [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC\|"BSC channel"]]  the received word   $\underline{y} = \underline{x} \, ({\rm code\:word}) + \underline{e} \, ({\rm error\:vector})$   is also an element of  ${\rm GF}(2^n)$,  and it holds because of   $ \underline{x} · \boldsymbol {{\rm H} }^{\rm T} =
	\underline{0}$:		\underline{0}$:
	:$$\underline{s} = \underline{e} \cdot { \boldsymbol{\rm H}}^{\rm T} \hspace{0.05cm}.$$		:$$\underline{s} = \underline{e} \cdot { \boldsymbol{\rm H}}^{\rm T} \hspace{0.05cm}.$$

2022-09-28T12:24:40Z

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	{{quiz-Header\|Buchseite=Channel_Coding/Decoding_of_Linear_Block_Codes}}		{{quiz-Header\|Buchseite=Channel_Coding/Decoding_of_Linear_Block_Codes}}

	[[File:EN_KC_A_1_11.png\|right\|frame\|Syndrome and coset leaders <br>$($incomplete list$)  of  $\rm HC \ (7, \, 4, \, 3)$ ]]		[[File:EN_KC_A_1_11.png\|right\|frame\|Syndrome and coset leaders <br>$($incomplete list$)$  of  $\rm HC \ (7, \, 4, \, 3)$ ]]

	To decode a  $(7, \, 4, \, 3)$  Hamming code,  which is defined by its parity-check matrix		To decode a  $(7, \, 4, \, 3)$  Hamming code,  which is defined by its parity-check matrix

2022-09-28T12:23:41Z

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	{{quiz-Header\|Buchseite=Channel_Coding/Decoding_of_Linear_Block_Codes}}		{{quiz-Header\|Buchseite=Channel_Coding/Decoding_of_Linear_Block_Codes}}

	[[File:EN_KC_A_1_11.png\|right\|frame\|Syndrome and coset leaders (incomplete list) of  $\rm HC \ (7, \, 4, \, 3)$ ]]		[[File:EN_KC_A_1_11.png\|right\|frame\|Syndrome and coset leaders <br>$($incomplete list$)  of  $\rm HC \ (7, \, 4, \, 3)$ ]]

	To decode a  $(7, \, 4, \, 3)$  Hamming code,  which is defined by its parity-check matrix		To decode a  $(7, \, 4, \, 3)$  Hamming code,  which is defined by its parity-check matrix

2022-08-17T14:23:38Z

← Older revision		Revision as of 16:23, 17 August 2022
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	{{quiz-Header\|Buchseite=Channel_Coding/Decoding_of_Linear_Block_Codes}}		{{quiz-Header\|Buchseite=Channel_Coding/Decoding_of_Linear_Block_Codes}}

	[[File:~~P_ID2397__KC_A_1_11~~.png\|right\|frame\|Syndrome and coset leaders (incomplete list) of  $\rm HC \ (7, \, 4, \, 3)$ ~~'''Korrektur'''~~ ]]		[[File:EN_KC_A_1_11.png\|right\|frame\|Syndrome and coset leaders (incomplete list) of  $\rm HC \ (7, \, 4, \, 3)$ ]]

	To decode a  $(7, \, 4, \, 3)$  Hamming code,  which is defined by its parity-check matrix		To decode a  $(7, \, 4, \, 3)$  Hamming code,  which is defined by its parity-check matrix