Aufgaben:Exercise 2.10Z: Code Rate and Minimum Distance - Revision history

Maintenance script: Fix interlanguage link: resolve redirect chain

2026-03-16T15:53:39Z

Fix interlanguage link: resolve redirect chain

← Older revision		Revision as of 17:53, 16 March 2026
Line 97:		Line 97:

	[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]		[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]
	[[de:Aufgaben:2.~~10Z_Coderate_und_minimale_Distanz~~]]		[[de:Aufgaben:Aufgabe 2.10Z: Coderate und minimale Distanz]]

Maintenance script: Add German interlanguage link

2026-03-16T12:30:49Z

Add German interlanguage link

← Older revision		Revision as of 14:30, 16 March 2026
Line 97:		Line 97:

	[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]		[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]
			[[de:Aufgaben:2.10Z_Coderate_und_minimale_Distanz]]

Hwang at 15:28, 23 January 2023

2023-01-23T15:28:47Z

← Older revision		Revision as of 17:28, 23 January 2023
Line 82:		Line 82:

	'''(3)'''  From  $d_{\rm min} = 33$  follows again  $t = 16 \ \Rightarrow \ N_{3} \ \underline{= 16}$.		'''(3)'''  From  $d_{\rm min} = 33$  follows again  $t = 16 \ \Rightarrow \ N_{3} \ \underline{= 16}$.
	*If exactly one bit is ~~corrupted~~ in each code symbol,  this also means  $16$  symbol errors.		*If exactly one bit is falsified in each code symbol,  this also means  $16$  symbol errors.

	*This is the maximum value that the Reed–Solomon decoder can still handle.		*This is the maximum value that the Reed–Solomon decoder can still handle.
Line 88:		Line 88:


	'''(4)'''  The Reed–Solomon decoder can correct  $16$  ~~corrupted~~ code symbols.		'''(4)'''  The Reed–Solomon decoder can correct  $16$  falsified code symbols.
	*It does not matter whether in a code symbol only one bit or all  $m = 8$  bits have been ~~corrupted~~.		*It does not matter whether in a code symbol only one bit or all  $m = 8$  bits have been falsified.

	*Therefore,  with the most favorable error distribution,  up to  $N_4 = 8 \cdot 16 \ \underline{= 128}$  bits can be ~~corrupted~~ without the code word being incorrectly decoded.		*Therefore,  with the most favorable error distribution,  up to  $N_4 = 8 \cdot 16 \ \underline{= 128}$  bits can be falsified without the code word being incorrectly decoded.
	{{ML-Fuß}}		{{ML-Fuß}}

Hwang at 19:03, 1 November 2022

2022-11-01T19:03:22Z

← Older revision		Revision as of 21:03, 1 November 2022
Line 12:		Line 12:
	* $k$  indicates the  "dimension"  $($symbol number of an information block$)$,		* $k$  indicates the  "dimension"  $($symbol number of an information block$)$,

	* $d_{\rm min}$  denotes the  "minimum distance"  between two ~~codewords~~.		* $d_{\rm min}$  denotes the  "minimum distance"  between two code words.

	*For any Reed-Solomon code:		*For any Reed-Solomon code:

Guenter at 10:43, 12 October 2022

2022-10-12T10:43:27Z

← Older revision		Revision as of 12:43, 12 October 2022
Line 33:		Line 33:
	===Questions===		===Questions===
	<quiz display=simple>		<quiz display=simple>
	{Specify the characteristics of the   ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_q$~~ ~~.		{Specify the characteristics of the   ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_q$.
	\|type="{}"}		\|type="{}"}
	$q \hspace{0.2cm} = \ ${ 256 }		$q \hspace{0.2cm} = \ ${ 256 }
Line 57:		Line 57:
	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  From the code length $n = 255$ follows $q \ \underline{= 256}$.		'''(1)'''  From the code length  $n = 255$  follows  $q \ \underline{= 256}$.

	*The code rate is given by $R = {223}/{255} \hspace{0.15cm}\underline {=0.8745}\hspace{0.05cm}.$		*The code rate is given by  $R = {223}/{255} \hspace{0.15cm}\underline {=0.8745}\hspace{0.05cm}.$

	*The minimum distance is $d_{\rm min} = n - k +1 = 255 - 223 +1		*The minimum distance is  $d_{\rm min} = n - k +1 = 255 - 223 +1
	\hspace{0.15cm}\underline {=33}\hspace{0.05cm}.$		\hspace{0.15cm}\underline {=33}\hspace{0.05cm}.$

	*This allows		*This allows:
	:* $e = d_{\rm min} - 1 \ \underline{= 32}$ symbol errors can be detected, and.		:* $e = d_{\rm min} - 1 \ \underline{= 32}$  symbol errors can be detected, and
	:* $t = e/2$ (rounded down)~~, so~~ $\underline{t = 16}$ symbol errors can be corrected.
			:* $t = e/2$  $($rounded down$)$.  So  $\underline{t = 16}$  symbol errors can be corrected.



	'''(2)'''  The code $\rm RSC \, (2040, \, 1784, \, d_{\rm min})_2$ is the binary representation of the ${\rm RSC} ~~discussed in (1)~~ \, (255, \, 223, \, 33)_{256}$ with exactly the same code rate $R \ \underline{= 0.8745}$ and ~~also the same minimum distance $d_{\rm min} \ \underline{= 33}$ as this one. Here $8$ bits (1 byte) are used per code symbol.~~		'''(2)'''  The code  $\rm RSC \, (2040, \, 1784, \, d_{\rm min})_2$  is the binary representation of the  ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_{256}$  discussed in  '''(1)'''
			*with exactly the same code rate  $R \ \underline{= 0.8745}$  and

			*also the same minimum distance  $d_{\rm min} \ \underline{= 33}$  as this one.


	'''(3)'''  From $d_{\rm min} = 33$ follows again $t = 16 \ \Rightarrow \ N_{3} \ \underline{= 16}$.		Here  $8$  bits  (1 byte)  are used per code symbol.
	*If exactly one bit is corrupted in each code symbol, this also means 16 symbol errors.


			'''(3)'''  From  $d_{\rm min} = 33$  follows again  $t = 16 \ \Rightarrow \ N_{3} \ \underline{= 16}$.
			*If exactly one bit is corrupted in each code symbol,  this also means  $16$  symbol errors.

	*This is the maximum value that the Reed–Solomon decoder can still handle.		*This is the maximum value that the Reed–Solomon decoder can still handle.


	'''(4)'''  The RS decoder can correct 16 corrupted code symbols,
	*~~whereby it~~ does not matter whether in a code symbol only one bit or all $m = 8$ bits have been corrupted.		'''(4)'''  The Reed–Solomon decoder can correct  $16$  corrupted code symbols.
	*Therefore, with the most favorable error distribution, up to $N_4 = 8 \cdot 16 \ \underline{= 128}$ bits can be corrupted without the code word being incorrectly decoded.		*It does not matter whether in a code symbol only one bit or all  $m = 8$  bits have been corrupted.

			*Therefore,  with the most favorable error distribution,  up to  $N_4 = 8 \cdot 16 \ \underline{= 128}$  bits can be corrupted without the code word being incorrectly decoded.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 10:03, 12 October 2022

2022-10-12T10:03:40Z

← Older revision		Revision as of 12:03, 12 October 2022
Line 23:		Line 23:

	Hints:		Hints:
	* The exercise belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and ~~properties~~ of Reed–Solomon Codes"]].		* The exercise belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and Properties of Reed–Solomon Codes"]].

	* Information relevant to this exercise can be found on the  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate\|"Code name and code rate"]] page.		* Information relevant to this exercise can be found on the  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate\|"Code name and code rate"]] page.

Guenter at 10:02, 12 October 2022

2022-10-12T10:02:53Z

← Older revision		Revision as of 12:02, 12 October 2022
Line 2:		Line 2:

	[[File:P_ID2526__KC_Z_2_10.png\|right\|frame\|The inventors of the Reed-Solomon codes]]		[[File:P_ID2526__KC_Z_2_10.png\|right\|frame\|The inventors of the Reed-Solomon codes]]
	The codes developed by  [https://en.wikipedia.org/wiki/Irving_S._Reed "Irving Stoy Reed"]  and  [https://en.wikipedia.org/wiki/Gustave_Solomon "Gustave Solomon"]  in the early 1960s are referred to in this tutorial as follows:		The codes developed by  [https://en.wikipedia.org/wiki/Irving_S._Reed Irving Stoy Reed]  and  [https://en.wikipedia.org/wiki/Gustave_Solomon Gustave Solomon]  in the early 1960s are referred to in this tutorial as follows:
	:$${\rm RSC} \, (n, \, k, \, d_{\rm min}) _q.$$		:$${\rm RSC} \, (n, \, k, \, d_{\rm min}) _q.$$

Line 8:		Line 8:
	* $q = 2^m$  is an indication of the  "size"  of the Galois field   ⇒   ${\rm GF}(q)$,		* $q = 2^m$  is an indication of the  "size"  of the Galois field   ⇒   ${\rm GF}(q)$,

	* $n = q - 1$  is the  "code length"  (symbol number of a code word),		* $n = q - 1$  is the  "code length"  $($symbol number of a code word$)$,

	* $k$  indicates the  "dimension"  (symbol number of an information block),		* $k$  indicates the  "dimension"  $($symbol number of an information block$)$,

	* $d_{\rm min}$  denotes the  "minimum distance"  between two codewords.		* $d_{\rm min}$  denotes the  "minimum distance"  between two codewords.
Line 24:		Line 24:
	Hints:		Hints:
	* The exercise belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and properties of Reed–Solomon Codes"]].		* The exercise belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and properties of Reed–Solomon Codes"]].

	* Information relevant to this exercise can be found on the  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate\|"Code name and code rate"]] page.		* Information relevant to this exercise can be found on the  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate\|"Code name and code rate"]] page.

Line 32:		Line 33:
	===Questions===		===Questions===
	<quiz display=simple>		<quiz display=simple>
	{Specify the characteristics of the  ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_q$ .		{Specify the characteristics of the   ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_q$ .
	\|type="{}"}		\|type="{}"}
	$q \hspace{0.2cm} = \ ${ 256 }		$q \hspace{0.2cm} = \ ${ 256 }
Line 40:		Line 41:
	$d_{\rm min} \ = \ ${ 33 }		$d_{\rm min} \ = \ ${ 33 }

	{Specify the characteristics of the  $\rm RSC \, (2040, \, 1784, \, d_{\rm min})_2$ .		{Specify the characteristics of the   $\rm RSC \, (2040, \, 1784, \, d_{\rm min})_2$ .
	\|type="{}"}		\|type="{}"}
	$R \hspace{0.2cm} = \ ${ 0.8745 3% }		$R \hspace{0.2cm} = \ ${ 0.8745 3% }
	$d_{\rm min} \ = \ ${ 33 }		$d_{\rm min} \ = \ ${ 33 }

	{How many bit errors  $(N_3)$  may a received word  $\underline{y}$  have at most, so that it is <u>certainly decoded correctly</u>?		{How many bit errors  $(N_3)$  may a received word  $\underline{y}$  have at most,  so that it is  <u>certainly decoded correctly</u>?
	\|type="{}"}		\|type="{}"}
	$N_{3} \ = \ $ { 16 }		$N_{3} \ = \ $ { 16 }

	{How many bit errors  $(N_4)$  may a received word  $\underline{y}$  have in the best case so that it could still be <u>correctly decoded</u>?		{How many bit errors  $(N_4)$  may a received word  $\underline{y}$  have  <u>in the best case</u>  so that it could still be  <u>correctly decoded</u>?
	\|type="{}"}		\|type="{}"}
	$N_{4} \ = \ $ { 128 }		$N_{4} \ = \ $ { 128 }

Guenter at 15:40, 11 October 2022

2022-10-11T15:40:52Z

← Older revision		Revision as of 17:40, 11 October 2022
Line 6:		Line 6:

	The code parameters have the following meanings:		The code parameters have the following meanings:
	* $q = 2^m$  is an indication of the size of the Galois field   ⇒   ${\rm GF}(q)$,		* $q = 2^m$  is an indication of the  "size"  of the Galois field   ⇒   ${\rm GF}(q)$,
	* $n = q - 1$  is the code length (symbol number of a code word),
	* $k$  indicates the dimension (symbol number of an information block),
	* $d_{\rm min}$  denotes the minimum distance between two codewords. For any Reed-Solomon code, $d_{\rm min} = n - k + 1$.
	*No other code with the same  $k$  and  $n$  yields a larger value.

			* $n = q - 1$  is the  "code length"  (symbol number of a code word),

			* $k$  indicates the  "dimension"  (symbol number of an information block),

			* $d_{\rm min}$  denotes the  "minimum distance"  between two codewords.

			*For any Reed-Solomon code:
			:$$d_{\rm min} = n - k + 1.$$

			'''No other code with the same  $k$  and  $n$  yields a larger value'''.

Guenter at 15:22, 4 September 2022

2022-09-04T15:22:39Z

← Older revision		Revision as of 17:22, 4 September 2022
Line 1:		Line 1:
	{{quiz-Header\|Buchseite=Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes}}		{{quiz-Header\|Buchseite=Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes}}

	[[File:P_ID2526__KC_Z_2_10.png\|right\|frame\|The ~~two~~ inventors of the Reed-Solomon codes]]		[[File:P_ID2526__KC_Z_2_10.png\|right\|frame\|The inventors of the Reed-Solomon codes]]
	The codes developed by  [https://en.wikipedia.org/wiki/Irving_S._Reed "Irving Stoy Reed"]  and  [https://en.wikipedia.org/wiki/Gustave_Solomon "Gustave Solomon"]  in the early 1960s are referred to in this tutorial as follows:		The codes developed by  [https://en.wikipedia.org/wiki/Irving_S._Reed "Irving Stoy Reed"]  and  [https://en.wikipedia.org/wiki/Gustave_Solomon "Gustave Solomon"]  in the early 1960s are referred to in this tutorial as follows:
	:$${\rm RSC} \, (n, \, k, \, d_{\rm min}) _q.$$		:$${\rm RSC} \, (n, \, k, \, d_{\rm min}) _q.$$

Noah at 18:19, 2 September 2022

2022-09-02T18:19:25Z

← Older revision		Revision as of 20:19, 2 September 2022
Line 21:		Line 21:
	Hints:		Hints:
	* The exercise belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and properties of Reed–Solomon Codes"]].		* The exercise belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and properties of Reed–Solomon Codes"]].
	* Information relevant to this exercise can be found on the  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate\|Code name and code rate]] page.		* Information relevant to this exercise can be found on the  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate\|"Code name and code rate"]] page.