Aufgaben:Exercise 2.09: Reed–Solomon Parameters - Revision history

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	[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]		[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]
	[[de:Aufgaben:2.~~09_Reed–Solomon–Parameter~~]]		[[de:Aufgaben:Aufgabe 2.09: Reed–Solomon–Parameter]]

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	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  For the code length of Reed–Solomon codes,  the following applies in general:		'''(1)'''  For the code length of Reed–Solomon codes,  the following applies in general:
	:$$n = q -1 = 2^m -1 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} m = 4 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 15} \hspace{0.05cm},		:$$n = q -1 = 2^m -1 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} m = 4 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 15} \hspace{0.05cm},\hspace{0.4cm}m = 5 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 31} \hspace{0.05cm},\hspace{0.4cm}m = 6 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 63} \hspace{0.05cm}. $$
	\hspace{0.4cm}m = 5 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 31} \hspace{0.05cm},\hspace{0.4cm}
	m = 6 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 63} \hspace{0.05cm}. $$

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	[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]		[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]
			[[de:Aufgaben:2.09_Reed–Solomon–Parameter]]

Guenter at 15:40, 10 October 2022

2022-10-10T15:40:22Z

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	===~~Solution~~===		===Questions===
	<quiz display=simple>		<quiz display=simple>
	{It holds  $c_i ∈ {\rm GF}(2^m)$.  Which Reed-Solomon code parameters  $n$  result?		{It holds  $c_i ∈ {\rm GF}(2^m)$.  Which Reed-Solomon code parameters  $n$  result?
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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  For the code length of Reed–Solomon codes, the following applies in general.		'''(1)'''  For the code length of Reed–Solomon codes,  the following applies in general:
	:$$n = q -1 = 2^m -1 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} m = 4 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 15} \hspace{0.05cm},		:$$n = q -1 = 2^m -1 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} m = 4 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 15} \hspace{0.05cm},
	\hspace{0.4cm}m = 5 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 31} \hspace{0.05cm},\hspace{0.4cm}		\hspace{0.4cm}m = 5 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 31} \hspace{0.05cm},\hspace{0.4cm}
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	'''(2)'''  To be able to correct $t$ symbol errors, the minimum distance must be $d_{\rm min} = 2t + 1$.		'''(2)'''  To be able to correct  $t$  symbol errors,  the minimum distance must be  $d_{\rm min} = 2t + 1$.
	*The Reed–Solomon code is a so-called ~~MDS code (<i>~~Maximum Distance Separable~~</i>~~).		*The Reed–Solomon code is a so-called  "Maximum Distance Separable  $\rm (MDS)$  code".

	*For this applies:		*For this applies:
	:$$d_{\rm min} = n-k+1 = 2t+1 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}k = n -2t = 2^m - ( 2t+1) \hspace{0.05cm}. $$		:$$d_{\rm min} = n-k+1 = 2t+1 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}k = n -2t = 2^m - ( 2t+1) \hspace{0.05cm}. $$

	*This gives for the		*This gives for the
	:* $\rm RSC \ 1$ (~~mit~~ $m = 4, \ t = 4) \text{:} \hspace{0.2cm} k = 2^4 - (2 \cdot 4 + 1) \ \underline{= 7}$,		** $\rm RSC \ 1$  $($with  $m = 4, \ t = 4) \text{:} \hspace{0.2cm} k = 2^4 - (2 \cdot 4 + 1) \ \underline{= 7}$,
	:* $\rm RSC \ 2$ (~~mit~~ $m = 5, \ t = 8) \text{:} \hspace{0.2cm} k = 2^5 - (2 \cdot 8 + 1) \ \underline{= 15}$.		** $\rm RSC \ 2$  $($with  $m = 5, \ t = 8) \text{:} \hspace{0.2cm} k = 2^5 - (2 \cdot 8 + 1) \ \underline{= 15}$.



	'''(3)'''  The denotation of a Reed–Solomon code is ${\rm RSC} \, (n, \, k, \, d_{\rm min})_q$ with $q = 2^m = n + 1$.		'''(3)'''  The denotation of a Reed–Solomon code is  ${\rm RSC} \, (n, \, k, \, d_{\rm min})_q$  with  $q = 2^m = n + 1$.

	Correct are the <u>solutions 1 and 4</u>:		*Correct are the  <u>solutions 1 and 4</u>:
	* $\rm RSC \ 1 \Rightarrow RSC \, (15, \, 7, \, 9)_{16}$,		** $\rm RSC \ 1 \Rightarrow RSC \, (15, \, 7, \, 9)_{16}$,
	* $\rm RSC \ 2 \Rightarrow RSC \, (31, \, 15, \, 17)_{32}$.		** $\rm RSC \ 2 \Rightarrow RSC \, (31, \, 15, \, 17)_{32}$.



	'''(4)'''  If $d_{\rm min}$ denotes the minimum distance of a block code, it can be used to detect $e = d_{\rm min} - 1$ symbol errors and correct $t = e/2$ symbol errors:		'''(4)'''  If  $d_{\rm min}$  denotes the minimum distance of a block code,  it can be used to detect  $e = d_{\rm min} - 1$  symbol errors and to correct  $t = e/2$  symbol errors:
	* ${\rm RSC} \ 1 \text{:} \hspace{0.2cm} d_{\rm min} = \ \ 9, \ t = 4, \ \underline{e = 8}$,		* ${\rm RSC} \ 1 \text{:} \hspace{0.2cm} d_{\rm min} = \ \ 9, \ t = 4, \ \underline{e = 8}$,
	* ${\rm RSC} \ 2 \text{:} \hspace{0.2cm} d_{\rm min} = 17, \ t = 8, \ \underline{e = 16}$.		* ${\rm RSC} \ 2 \text{:} \hspace{0.2cm} d_{\rm min} = 17, \ t = 8, \ \underline{e = 16}$.
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	'''(5)'''  Correct are the two middle <u>solutions 2 and 3</u>:		'''(5)'''  Correct are the two middle  <u>solutions 2 and 3</u>:
	*For ${\rm RSC} \ 1 \, (m = 4)$, $n = 15$ code symbols from $\rm GF(2^5)$ ~~equal~~ $60$ bits and $k = 7$ information symbols ~~equal~~ exactly $28$ bits:		*For  ${\rm RSC} \ 1 \, (m = 4)$:   $n = 15$  code symbols from  $\rm GF(2^5)$  correspond to  $60$  bits and  $k = 7$  information symbols are exactly  $28$  bits:
	:* $\rm RSC \ 1 \Rightarrow RSC \, (15, \, 7, \, 9)_{16} \Rightarrow RSC \, (60, \, 28, \, 9)_2$,		** $\rm RSC \ 1 \Rightarrow RSC \, (15, \, 7, \, 9)_{16} \Rightarrow RSC \, (60, \, 28, \, 9)_2$,
	:* $\rm RSC \ 2 \Rightarrow RSC \, (31, \, 15, \, 17)_{32} \Rightarrow RSC \, (155, \, 75, \, 17)_2$.		** $\rm RSC \ 2 \Rightarrow RSC \, (31, \, 15, \, 17)_{32} \Rightarrow RSC \, (155, \, 75, \, 17)_2$.
	*For the minimum distance on bit level   ⇒   $\rm GF(2)$, $d_{\rm min} = 9$ resp. $d_{\rm min} = 17$ result in the same values as on symbol level   ~~<br>⇒   $\rm GF(2^4)~~$ ~~resp. $\rm GF~~(~~2^5)~~$ (see [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate\|"theory section"]]).		*For the minimum distance on bit level   ⇒   $\rm GF(2)$,  $d_{\rm min} = 9$  resp.  $d_{\rm min} = 17$  result in the same values as on symbol level   $($see [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate\|"theory section"]]$)$.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 15:17, 10 October 2022

2022-10-10T15:17:28Z

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	[[File:P_ID2523__KC_A_2_9_neu.png\|right\|frame\|Some Reed-Solomon codes]]		[[File:P_ID2523__KC_A_2_9_neu.png\|right\|frame\|Some Reed-Solomon codes]]
	Adjacent is an incomplete list of possible Reed–Solomon codes known to be based on a Galois field  ${\rm GF}(q) = {\rm GF}(2^m)$ . The parameter  $m$  specifies with how many bits a RS code symbol is represented. It is valid:		Adjacent is an incomplete list of possible Reed–Solomon codes known to be based on a Galois field  ${\rm GF}(q) = {\rm GF}(2^m)$.

			The parameter  $m$  specifies with how many bits a Reed–Solomon code symbol is represented.  It is valid:
	* $m = 4$  (red font),		* $m = 4$  (red font),

	* $m = 5$  (blue font),		* $m = 5$  (blue font),

	* $m = 6$  (green font).		* $m = 6$  (green font).


	A Reed–Solomon code is generally denoted as follows:   ${\rm RSC}\ (n, \ k, \ d_{\rm min})_q$.		A Reed–Solomon code is generally denoted as follows:   ${\rm RSC}\ (n, \ k, \ d_{\rm min})_q$.


	The parameters have the following meaning:		The parameters have the following meaning:
	* $n$  specifies the number of symbols of a ~~codeword~~  $\underline{c}$    ⇒   <b>length</b> of the code,		# The parameter  $n$  specifies the number of symbols of a code word  $\underline{c}$    ⇒   <b>length</b> of the code.
	* $k$  specifies the number of symbols of an information block  $\underline{u}$    ⇒   <b>dimension</b> of the code,		# Tthe parameter  $k$  specifies the number of symbols of an information block  $\underline{u}$    ⇒   <b>dimension</b>  of the code.
	* $d_{\rm min}$  denotes the  <b> minimum distance </b>  between two ~~codewords~~ <br>$($for all Reed–Solomon codes equal  $n-k+1)$,		# The parameter  $d_{\rm min}$  denotes the  <b> minimum distance </b>  between two code words <br>$($for all Reed–Solomon codes equal  $n-k+1)$.
	* $q$  gives an indication of the use of the Galois field  ${\rm GF}(q)$.		# The parameter  $q$  gives an indication of the use of the Galois field  ${\rm GF}(q)$.


	~~To the right is the binary representation of the same code:~~
	*In this realization of a RS code, each information and code symbol is represented by  $m$  bits.
	*For example, it can be seen from the first row that the minimum distance in terms of bits is also  $d_{\rm min} = 5$  if in  ${\rm GF}(2^m)$  the minimum distance is  $d_{\rm min} = 5$ .
	*This can correct up to  $t = 2$  bit errors (or symbol errors) and detect up to  $e = 4$  bit errors (or symbol errors).


			To the right,  there is the binary representation of the same code:
			# In this realization of a Reed–Solomon code,  each information and code symbol is represented by  $m$  bits.
			# For example,  it can be seen from the first row that the minimum distance in terms of bits is also  $d_{\rm min} = 5$  if in  ${\rm GF}(2^m)$  the minimum distance is  $d_{\rm min} = 5$.
			# This code can correct up to   $t = 2$   bit errors  $($or symbol errors$)$  and detect up to   $e = 4$   bit errors  $($or symbol errors$)$.


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	Hints:		Hints:
	* This exercise belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and Properties of Reed-Solomon Codes"]].		* This exercise belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and Properties of Reed-Solomon Codes"]].
	* However, reference is also made to the chapter  [[Channel_Coding/Extension_Field\| "Extension Field"]].
			* However,  reference is also made to the chapter  [[Channel_Coding/Extension_Field\| "Extension Field"]].


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	===Solution===		===Solution===
	<quiz display=simple>		<quiz display=simple>
	{It ~~is valid~~  $c_i ∈ {\rm GF}(2^m)$. Which RS code parameters  $n$  result?		{It holds  $c_i ∈ {\rm GF}(2^m)$.  Which Reed-Solomon code parameters  $n$  result?
	\|type="{}"}		\|type="{}"}
	$m = 4 \text{:} \hspace{0.4cm} n \ = \ ${ 15 }		$m = 4 \text{:} \hspace{0.4cm} n \ = \ ${ 15 }
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	$m = 6 \text{:} \hspace{0.4cm} n \ = \ ${ 63 }		$m = 6 \text{:} \hspace{0.4cm} n \ = \ ${ 63 }

	{Two specific RS codes  ${\rm RSC} \ 1 \ (m = 4, \ t = 4)$  and  ${\rm RSC} \ 2 \ (m = 5, \ t = 8)$  are considered below. <br>Which RS parameter  $k$  can be used to correct exactly  $t$  symbol errors?		{Two specific Reed-Solomon codes   ${\rm RSC} \ 1 \ (m = 4, \ t = 4)$   and   ${\rm RSC} \ 2 \ (m = 5, \ t = 8)$   are considered below. <br>Which parameter  $k$  can be used to correct exactly  $t$  symbol errors?
	\|type="{}"}		\|type="{}"}
	${\rm RSC} \ 1 \text{:} \hspace{0.4cm} k \ = \ ${ 7 }		${\rm RSC} \ 1 \text{:} \hspace{0.4cm} k \ = \ ${ 7 }
	${\rm RSC} \ 2 \text{:} \hspace{0.4cm} k \ = \ ${ 15 }		${\rm RSC} \ 2 \text{:} \hspace{0.4cm} k \ = \ ${ 15 }

	{Which designations are correct for  $\rm RSC \ 1$  ~~and~~  $\rm RSC \ 2$~~  respectively~~?		{Which designations are correct for   $\rm RSC \ 1$   resp.   $\rm RSC \ 2$?
	\|type="[]"}		\|type="[]"}
	+ $\rm RSC \ 1$  ~~nennt man auch~~  $\rm RSC \, (15, \, 7, \, 9)_{16}$.		+ $\rm RSC \ 1$  is also called  $\rm RSC \, (15, \, 7, \, 9)_{16}$.
	- $\rm RSC \ 1$  ~~nennt man auch~~  $\rm RSC \, (15, \, 7, \, 4)_4$.		- $\rm RSC \ 1$  is also called  $\rm RSC \, (15, \, 7, \, 4)_4$.
	- $\rm RSC \ 2$  ~~nennt man auch~~  $\rm RSC \, (31, \, 17, \, 15)_{32}$.		- $\rm RSC \ 2$  is also called  $\rm RSC \, (31, \, 17, \, 15)_{32}$.
	+ $\rm RSC \ 2$  ~~nennt man auch~~  $\rm RSC \, (31, \, 15, \, 17)_{32}$.		+ $\rm RSC \ 2$  is also called  $\rm RSC \, (31, \, 15, \, 17)_{32}$.

	{How many symbol errors  $(e)$  can be detected at most?		{How many symbol errors  $(e)$  can be detected at most?
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	{What are the codes under consideration in binary notation?		{What are the codes under consideration in binary notation?
	\|type="[]"}		\|type="[]"}
	- $\rm RSC \ 1$  corresponds to code  $\rm RSC \, (60, \, 28, \, 36)_2$.		- $\rm RSC \ 1$  corresponds to the code  $\rm RSC \, (60, \, 28, \, 36)_2$.
	+ $\rm RSC \ 1$  corresponds to code  $\rm RSC \, (60, \, 28, \, 9)_2$.		+ $\rm RSC \ 1$  corresponds to the code  $\rm RSC \, (60, \, 28, \, 9)_2$.
	+ $\rm RSC \ 2$  corresponds to code  $\rm RSC \, (155, \, 75, \, 17)_2$.		+ $\rm RSC \ 2$  corresponds to the code  $\rm RSC \, (155, \, 75, \, 17)_2$.
	- $\rm RSC \ 2$  corresponds to code  $\rm RSC \, (124, \, 60, \, 17)_2$.		- $\rm RSC \ 2$  corresponds to the code  $\rm RSC \, (124, \, 60, \, 17)_2$.
	</quiz>		</quiz>

Noah at 16:49, 2 September 2022

2022-09-02T16:49:35Z

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	Hints:		Hints:
	* ~~Die Aufgabe gehört zum Kapitel~~  [[Channel_Coding/~~Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes~~\| Definition ~~und Eigenschaften von~~ Reed~~–~~Solomon~~–~~Codes]].		* This exercise belongs to the chapter  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes\| "Definition and Properties of Reed-Solomon Codes"]].
	* ~~Bezug genommen wird aber auch auf das Kapitel~~  [[Channel_Coding/~~Erweiterungsk%C3%B6rper~~\| ~~Erweiterungskörper~~]].		* However, reference is also made to the chapter  [[Channel_Coding/Extension_Field\| "Extension Field"]].




	===~~Fragebogen~~===		===Solution===
	<quiz display=simple>		<quiz display=simple>
	{~~Es gelte~~  $c_i ∈ {\rm GF}(2^m)$. ~~Welche~~ RS~~–Codeparameter~~  $n$  ~~ergeben sich~~?		{It is valid  $c_i ∈ {\rm GF}(2^m)$. Which RS code parameters  $n$  result?
	\|type="{}"}		\|type="{}"}
	$m = 4 \text{:} \hspace{0.4cm} n \ = \ ${ 15 }		$m = 4 \text{:} \hspace{0.4cm} n \ = \ ${ 15 }
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	$m = 6 \text{:} \hspace{0.4cm} n \ = \ ${ 63 }		$m = 6 \text{:} \hspace{0.4cm} n \ = \ ${ 63 }

	{~~Im Folgenden werden zwei spezielle~~ RS~~–Codes~~  ${\rm RSC} \ 1 \ (m = 4, \ t = 4)$  ~~und~~  ${\rm RSC} \ 2 \ (m = 5, \ t = 8)$  ~~betrachtet~~. <br>~~Mit welchem~~ RS~~–Parameter~~  $k$  ~~lassen sich genau~~  $t$  ~~Symbolfehler korrigeren~~?		{Two specific RS codes  ${\rm RSC} \ 1 \ (m = 4, \ t = 4)$  and  ${\rm RSC} \ 2 \ (m = 5, \ t = 8)$  are considered below. <br>Which RS parameter  $k$  can be used to correct exactly  $t$  symbol errors?
	\|type="{}"}		\|type="{}"}
	${\rm RSC} \ 1 \text{:} \hspace{0.4cm} k \ = \ ${ 7 }		${\rm RSC} \ 1 \text{:} \hspace{0.4cm} k \ = \ ${ 7 }
	${\rm RSC} \ 2 \text{:} \hspace{0.4cm} k \ = \ ${ 15 }		${\rm RSC} \ 2 \text{:} \hspace{0.4cm} k \ = \ ${ 15 }

	{~~Welche Bezeichnungen sind für~~  $\rm RSC \ 1$  ~~bzw.~~  $\rm RSC \ 2$  ~~richtig~~?		{Which designations are correct for  $\rm RSC \ 1$  and  $\rm RSC \ 2$  respectively?
	\|type="[]"}		\|type="[]"}
	+ $\rm RSC \ 1$  nennt man auch  $\rm RSC \, (15, \, 7, \, 9)_{16}$.		+ $\rm RSC \ 1$  nennt man auch  $\rm RSC \, (15, \, 7, \, 9)_{16}$.
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	+ $\rm RSC \ 2$  nennt man auch  $\rm RSC \, (31, \, 15, \, 17)_{32}$.		+ $\rm RSC \ 2$  nennt man auch  $\rm RSC \, (31, \, 15, \, 17)_{32}$.

	{~~Wieviele Symbolfehler~~  $(e)$  ~~können höchstens erkannt werden~~?		{How many symbol errors  $(e)$  can be detected at most?
	\|type="{}"}		\|type="{}"}
	${\rm RSC} \ 1 \text{:} \hspace{0.4cm} e \ = \ ${ 8 }		${\rm RSC} \ 1 \text{:} \hspace{0.4cm} e \ = \ ${ 8 }
	${\rm RSC} \ 2 \text{:} \hspace{0.4cm}e \ = \ ${ 16 }		${\rm RSC} \ 2 \text{:} \hspace{0.4cm}e \ = \ ${ 16 }

	{~~Wie lauten die betrachteten Codes~~ in ~~Binärschreibweise~~?		{What are the codes under consideration in binary notation?
	\|type="[]"}		\|type="[]"}
	- $\rm RSC \ 1$  ~~entspricht dem Code~~  $\rm RSC \, (60, \, 28, \, 36)_2$.		- $\rm RSC \ 1$  corresponds to code  $\rm RSC \, (60, \, 28, \, 36)_2$.
	+ $\rm RSC \ 1$  ~~entspricht dem Code~~  $\rm RSC \, (60, \, 28, \, 9)_2$.		+ $\rm RSC \ 1$  corresponds to code  $\rm RSC \, (60, \, 28, \, 9)_2$.
	+ $\rm RSC \ 2$  ~~entspricht dem Code~~  $\rm RSC \, (155, \, 75, \, 17)_2$.		+ $\rm RSC \ 2$  corresponds to code  $\rm RSC \, (155, \, 75, \, 17)_2$.
	- $\rm RSC \ 2$  ~~entspricht dem Code~~  $\rm RSC \, (124, \, 60, \, 17)_2$.		- $\rm RSC \ 2$  corresponds to code  $\rm RSC \, (124, \, 60, \, 17)_2$.
	</quiz>		</quiz>

	===~~Musterlösung~~===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  ~~Für die Codelänge der~~ Reed–Solomon~~–Codes gilt allgemein~~.		'''(1)'''  For the code length of Reed–Solomon codes, the following applies in general.
	:$$n = q -1 = 2^m -1 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} m = 4 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 15} \hspace{0.05cm},		:$$n = q -1 = 2^m -1 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} m = 4 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 15} \hspace{0.05cm},
	\hspace{0.4cm}m = 5 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 31} \hspace{0.05cm},\hspace{0.4cm}		\hspace{0.4cm}m = 5 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 31} \hspace{0.05cm},\hspace{0.4cm}
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	'''(2)'''  Um $t$ ~~Symbolfehler korrigieren zu können~~, ~~muss die minimale Distanz~~ $d_{\rm min} = 2t + 1$ ~~betragen~~.		'''(2)'''  To be able to correct $t$ symbol errors, the minimum distance must be $d_{\rm min} = 2t + 1$.
	*~~Der~~ Reed–Solomon~~–Code ist ein sogenannter~~ MDS~~–Code~~ (<i>Maximum Distance Separable</i>).		*The Reed–Solomon code is a so-called MDS code (<i>Maximum Distance Separable</i>).
	*~~Für diese gilt~~:		*For this applies:
	:$$d_{\rm min} = n-k+1 = 2t+1 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}k = n -2t = 2^m - ( 2t+1) \hspace{0.05cm}. $$		:$$d_{\rm min} = n-k+1 = 2t+1 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}k = n -2t = 2^m - ( 2t+1) \hspace{0.05cm}. $$

	*~~Damit erhält man für den~~		*This gives for the
	:* $\rm RSC \ 1$ (mit $m = 4, \ t = 4) \text{:} \hspace{0.2cm} k = 2^4 - (2 \cdot 4 + 1) \ \underline{= 7}$,		:* $\rm RSC \ 1$ (mit $m = 4, \ t = 4) \text{:} \hspace{0.2cm} k = 2^4 - (2 \cdot 4 + 1) \ \underline{= 7}$,
	:* $\rm RSC \ 2$ (mit $m = 5, \ t = 8) \text{:} \hspace{0.2cm} k = 2^5 - (2 \cdot 8 + 1) \ \underline{= 15}$.		:* $\rm RSC \ 2$ (mit $m = 5, \ t = 8) \text{:} \hspace{0.2cm} k = 2^5 - (2 \cdot 8 + 1) \ \underline{= 15}$.
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	'''(3)'''  ~~Die Bezeichnung eines~~ Reed–Solomon~~–Codes lautet~~ ${\rm RSC} \, (n, \, k, \, d_{\rm min})_q$ ~~mit~~ $q = 2^m = n + 1$.		'''(3)'''  The denotation of a Reed–Solomon code is ${\rm RSC} \, (n, \, k, \, d_{\rm min})_q$ with $q = 2^m = n + 1$.

	~~Richtig sind die~~ <u>~~Lösungsvorschläge~~ 1 ~~und~~ 4</u>:		Correct are the <u>solutions 1 and 4</u>:
	* $\rm RSC \ 1 \Rightarrow RSC \, (15, \, 7, \, 9)_{16}$,		* $\rm RSC \ 1 \Rightarrow RSC \, (15, \, 7, \, 9)_{16}$,
	* $\rm RSC \ 2 \Rightarrow RSC \, (31, \, 15, \, 17)_{32}$.		* $\rm RSC \ 2 \Rightarrow RSC \, (31, \, 15, \, 17)_{32}$.
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	'''(4)'''  ~~Bezeichnet~~ $d_{\rm min}$ ~~die minimale Distanz eines Blockcodes~~, ~~so können damit~~ $e = d_{\rm min} - 1$ ~~Symbolfehler erkannt und~~ $t = e/2$ ~~Symbolfehler korrigiert werden~~:		'''(4)'''  If $d_{\rm min}$ denotes the minimum distance of a block code, it can be used to detect $e = d_{\rm min} - 1$ symbol errors and correct $t = e/2$ symbol errors:
	* ${\rm RSC} \ 1 \text{:} \hspace{0.2cm} d_{\rm min} = \ \ 9, \ t = 4, \ \underline{e = 8}$,		* ${\rm RSC} \ 1 \text{:} \hspace{0.2cm} d_{\rm min} = \ \ 9, \ t = 4, \ \underline{e = 8}$,
	* ${\rm RSC} \ 2 \text{:} \hspace{0.2cm} d_{\rm min} = 17, \ t = 8, \ \underline{e = 16}$.		* ${\rm RSC} \ 2 \text{:} \hspace{0.2cm} d_{\rm min} = 17, \ t = 8, \ \underline{e = 16}$.
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	'''(5)'''  ~~Richtig sind die beiden mittleren~~ <u>~~Lösungsvorschläge~~ 2 ~~und~~ 3</u>:		'''(5)'''  Correct are the two middle <u>solutions 2 and 3</u>:
	*~~Bei~~ ${\rm RSC} \ 1 \, (m = 4)$ ~~entsprechen~~ $n = 15$ ~~Codesymbolen aus~~ $\rm GF(2^5)$ ~~gleich~~ $60$ ~~Bit und~~ $k = 7$ ~~Informationssymbole genau~~ $28$ ~~Bit~~:		*For ${\rm RSC} \ 1 \, (m = 4)$, $n = 15$ code symbols from $\rm GF(2^5)$ equal $60$ bits and $k = 7$ information symbols equal exactly $28$ bits:
	:* $\rm RSC \ 1 \Rightarrow RSC \, (15, \, 7, \, 9)_{16} \Rightarrow RSC \, (60, \, 28, \, 9)_2$,		:* $\rm RSC \ 1 \Rightarrow RSC \, (15, \, 7, \, 9)_{16} \Rightarrow RSC \, (60, \, 28, \, 9)_2$,
	:* $\rm RSC \ 2 \Rightarrow RSC \, (31, \, 15, \, 17)_{32} \Rightarrow RSC \, (155, \, 75, \, 17)_2$.		:* $\rm RSC \ 2 \Rightarrow RSC \, (31, \, 15, \, 17)_{32} \Rightarrow RSC \, (155, \, 75, \, 17)_2$.
	*~~Für die minimale Distanz auf Bitebene~~   ⇒   $\rm GF(2)$ ~~ergeben sich mit~~ $d_{\rm min} = 9$ ~~bzw~~. $d_{\rm min} = 17$ ~~die gleichen Werte wie auf Symbolebene~~   <br>⇒   $\rm GF(2^4)$ ~~bzw~~. $\rm GF(2^5)$ (~~siehe~~ [[Channel_Coding/~~Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes~~#~~Codebezeichnung_und_Coderate~~\|~~Theorieteil~~]]).		*For the minimum distance on bit level   ⇒   $\rm GF(2)$, $d_{\rm min} = 9$ resp. $d_{\rm min} = 17$ result in the same values as on symbol level   <br>⇒   $\rm GF(2^4)$ resp. $\rm GF(2^5)$ (see [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate\|"theory section"]]).
	{{ML-Fuß}}		{{ML-Fuß}}

Noah at 16:40, 2 September 2022

2022-09-02T16:40:43Z

← Older revision		Revision as of 18:40, 2 September 2022
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	{{quiz-Header\|Buchseite=~~Kanalcodierung~~/~~Definition und Eigenschaften von Reed–Solomon–Codes~~}}		{{quiz-Header\|Buchseite=Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes}}

	[[File:P_ID2523__KC_A_2_9_neu.png\|right\|frame\|~~Einige Reed–Solomon–Codes~~]]		[[File:P_ID2523__KC_A_2_9_neu.png\|right\|frame\|Some Reed-Solomon codes]]
	~~Nebenstehend finden Sie eine unvollständige Liste möglicher~~ Reed–Solomon~~–Codes, die bekanntlich auf einem Galoisfeld~~  ${\rm GF}(q) = {\rm GF}(2^m)$  ~~basieren~~. ~~Der Parameter~~  $m$  ~~gibt an, mit wie vielen Bit ein~~ RS~~–Codesymbol dargestellt wird~~. ~~Es gilt~~:		Adjacent is an incomplete list of possible Reed–Solomon codes known to be based on a Galois field  ${\rm GF}(q) = {\rm GF}(2^m)$ . The parameter  $m$  specifies with how many bits a RS code symbol is represented. It is valid:
	* $m = 4$  (~~rote Schrift~~),		* $m = 4$  (red font),
	* $m = 5$  (~~blaue Schrift~~),		* $m = 5$  (blue font),
	* $m = 6$  (~~grüne Schrift~~).		* $m = 6$  (green font).


	~~Ein~~ Reed–Solomon~~–Code wird allgemein wie folgt bezeichnet~~:   ${\rm RSC}\ (n, \ k, \ d_{\rm min})_q$.		A Reed–Solomon code is generally denoted as follows:   ${\rm RSC}\ (n, \ k, \ d_{\rm min})_q$.


	~~Die Parameter haben folgende Bedeutung~~:		The parameters have the following meaning:
	* $n$  ~~gibt die Anzahl der Symbole eines Codewortes~~  $\underline{c}$  an   ⇒   <b>~~Länge~~</b> ~~des Codes~~,		* $n$  specifies the number of symbols of a codeword  $\underline{c}$    ⇒   <b>length</b> of the code,
	* $k$  ~~gibt die Anzahl der Symbole eines Informationsblocks~~  $\underline{u}$  an   ⇒   <b>~~Dimension~~</b> ~~des Codes~~,		* $k$  specifies the number of symbols of an information block  $\underline{u}$    ⇒   <b>dimension</b> of the code,
	* $d_{\rm min}$  ~~kennzeichnet die~~  <b>~~minimale Distanz~~ </b>  ~~zwischen zwei Codeworten~~ <br>$($~~bei allen~~ Reed–Solomon~~–Codes gleich~~  $n-k+1)$,		* $d_{\rm min}$  denotes the  <b> minimum distance </b>  between two codewords <br>$($for all Reed–Solomon codes equal  $n-k+1)$,
	* $q$  ~~gibt einen Hinweis auf die Verwendung des Galoisfeldes~~  ${\rm GF}(q)$.		* $q$  gives an indication of the use of the Galois field  ${\rm GF}(q)$.


	~~Rechts daneben ist die Binärrepräsentation des gleichen Codes angegeben~~:		To the right is the binary representation of the same code:
	*~~Bei dieser Realisierung eines~~ RS~~–Codes wird jedes Informations– und Codesymbol durch~~  $m$  ~~Bit dargestellt~~.		*In this realization of a RS code, each information and code symbol is represented by  $m$  bits.
	*~~Beispielsweise erkennt man aus der ersten Zeile~~, ~~dass die minimale Distanz hinsichtlich der Bits ebenfalls~~  $d_{\rm min} = 5$  ~~ist, wenn~~ in  ${\rm GF}(2^m)$  ~~die minimale Distanz~~  $d_{\rm min} = 5$  ~~beträgt~~.		*For example, it can be seen from the first row that the minimum distance in terms of bits is also  $d_{\rm min} = 5$  if in  ${\rm GF}(2^m)$  the minimum distance is  $d_{\rm min} = 5$ .
	*~~Damit können bis zu~~  $t = 2$  ~~Bitfehler~~ (~~oder Symbolfehler~~) ~~korrigiert und bis zu~~  $e = 4$  ~~Bitfehler~~ (~~oder Symbolfehler~~) ~~erkannt werden~~.		*This can correct up to  $t = 2$  bit errors (or symbol errors) and detect up to  $e = 4$  bit errors (or symbol errors).


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	~~''Hinweise~~:''		Hints:
	* Die Aufgabe gehört zum Kapitel  [[Channel_Coding/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes\| Definition und Eigenschaften von Reed–Solomon–Codes]].		* Die Aufgabe gehört zum Kapitel  [[Channel_Coding/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes\| Definition und Eigenschaften von Reed–Solomon–Codes]].
	* Bezug genommen wird aber auch auf das Kapitel  [[Channel_Coding/Erweiterungsk%C3%B6rper\| Erweiterungskörper]].		* Bezug genommen wird aber auch auf das Kapitel  [[Channel_Coding/Erweiterungsk%C3%B6rper\| Erweiterungskörper]].

Guenter at 12:31, 2 September 2022

2022-09-02T12:31:07Z

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	[[Category:Channel Coding: Exercises\|^2.3 ~~Zu den Reed–Solomon–Codes~~^]]		[[Category:Channel Coding: Exercises\|^2.3 Reed–Solomon Codes^]]

Guenter: Guenter moved page Aufgaben:Aufgabe 2.09: Reed–Solomon–Parameter to Aufgaben:Exercise 2.09: Reed–Solomon Parameters

2022-09-02T12:25:21Z

Guenter moved page Aufgaben:Aufgabe 2.09: Reed–Solomon–Parameter to Aufgaben:Exercise 2.09: Reed–Solomon Parameters

← Older revision	Revision as of 14:25, 2 September 2022
(No difference)

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

2021-03-23T12:48:22Z

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

← Older revision		Revision as of 14:48, 23 March 2021
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	[[Category:~~Aufgaben zu Kanalcodierung~~\|^2.3 Zu den Reed–Solomon–Codes^]]		[[Category:Channel Coding: Exercises\|^2.3 Zu den Reed–Solomon–Codes^]]