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

Guenter at 15:40, 10 October 2022

2022-10-10T15:40:22Z

Guenter at 15:17, 10 October 2022

2022-10-10T15:17:28Z

Noah at 16:49, 2 September 2022

2022-09-02T16:49:35Z

Noah at 16:40, 2 September 2022

2022-09-02T16:40:43Z

Guenter at 12:31, 2 September 2022

2022-09-02T12:31:07Z

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 Aufgabe 2.09: Reed–Solomon–Parameter to Exercise 2.09: Reed–Solomon Parameters

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"

Javier: Text replacement - "[[Kanalcodierung" to "[[Channel_Coding"

2020-07-09T12:44:43Z

Text replacement - "[[Kanalcodierung" to "[[Channel_Coding"

Guenter at 15:12, 22 May 2019

2019-05-22T15:12:42Z

Guenter at 15:07, 22 May 2019

2019-05-22T15:07:14Z

@@ Line 38: / Line 38: @@
-===Solution===
+===Questions===
 <quiz display=simple>
 {It holds&nbsp; $c_i &#8712; {\rm GF}(2^m)$.&nbsp; Which Reed-Solomon code parameters&nbsp; $n$&nbsp; result?
@@ Line 73: / Line 73: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; For the code length of Reed&ndash;Solomon codes, the following applies in general.
+'''(1)'''&nbsp; For the code length of Reed&ndash;Solomon codes,&nbsp; the following applies in general:
 :$$n = q -1 = 2^m -1 \hspace{0.3cm}\Rightarrow  \hspace{0.3cm} m = 4 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 15} \hspace{0.05cm},
 \hspace{0.4cm}m = 5 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 31} \hspace{0.05cm},\hspace{0.4cm}
@@ Line 79: / Line 79: @@
-'''(2)'''&nbsp; To be able to correct $t$ symbol errors, the minimum distance must be $d_{\rm min} = 2t + 1$.
+'''(2)'''&nbsp; To be able to correct&nbsp; $t$&nbsp; symbol errors,&nbsp; the minimum distance must be&nbsp; $d_{\rm min} = 2t + 1$.
-*The Reed&ndash;Solomon code is a so-called MDS code (<i>Maximum Distance Separable</i>).
+*The Reed&ndash;Solomon code is a so-called&nbsp; "Maximum Distance Separable&nbsp; $\rm (MDS)$&nbsp; code".
 *For this applies:
 :$$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}. $$
 *This gives for the
-:* $\rm RSC \ 1$ (mit $m = 4, \ t = 4) \text{:} \hspace{0.2cm} k = 2^4 - (2 \cdot 4 + 1) \ \underline{= 7}$,
+** $\rm RSC \ 1$&nbsp; $($with&nbsp; $m = 4, \ t = 4) \text{:} \hspace{0.2cm} k = 2^4 - (2 \cdot 4 + 1) \ \underline{= 7}$,
-:* $\rm RSC \ 2$ (mit $m = 5, \ t = 8) \text{:} \hspace{0.2cm} k = 2^5 - (2 \cdot 8 + 1) \ \underline{= 15}$.
+** $\rm RSC \ 2$&nbsp; $($with&nbsp; $m = 5, \ t = 8) \text{:} \hspace{0.2cm} k = 2^5 - (2 \cdot 8 + 1) \ \underline{= 15}$.
-'''(3)'''&nbsp; The denotation of a Reed&ndash;Solomon code is ${\rm RSC} \, (n, \, k, \, d_{\rm min})_q$ with $q = 2^m = n + 1$.
+'''(3)'''&nbsp; The denotation of a Reed&ndash;Solomon code is&nbsp; ${\rm RSC} \, (n, \, k, \, d_{\rm min})_q$&nbsp; with&nbsp; $q = 2^m = n + 1$.
-Correct are the <u>solutions 1 and 4</u>:
+*Correct are the&nbsp; <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)'''&nbsp; 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)'''&nbsp; If&nbsp; $d_{\rm min}$&nbsp; denotes the minimum distance of a block code,&nbsp; it can be used to detect&nbsp; $e = d_{\rm min} - 1$&nbsp; symbol errors and to correct&nbsp; $t = e/2$&nbsp; symbol errors:
 * ${\rm RSC} \ 1 \text{:} \hspace{0.2cm} d_{\rm min} = \ \ 9, \ t = 4, \ \underline{e = 8}$,
 * ${\rm RSC} \ 2 \text{:} \hspace{0.2cm} d_{\rm min} = 17, \ t = 8, \ \underline{e = 16}$.
@@ Line 104: / Line 105: @@
-'''(5)'''&nbsp; Correct are the two middle <u>solutions 2 and 3</u>:
+'''(5)'''&nbsp; Correct are the two middle&nbsp; <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&nbsp; ${\rm RSC} \ 1 \, (m = 4)$: &nbsp; $n = 15$&nbsp; code symbols from&nbsp; $\rm GF(2^5)$&nbsp; correspond to&nbsp; $60$&nbsp; bits and&nbsp; $k = 7$&nbsp; information symbols are exactly&nbsp; $28$&nbsp; 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 &nbsp; &rArr; &nbsp; $\rm GF(2)$, $d_{\rm min} = 9$ resp. $d_{\rm min} = 17$ result in the same values as on symbol level &nbsp; <br>&rArr; &nbsp; $\rm GF(2^4)$ 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 &nbsp; &rArr; &nbsp; $\rm GF(2)$,&nbsp; $d_{\rm min} = 9$&nbsp; resp.&nbsp; $d_{\rm min} = 17$&nbsp; result in the same values as on symbol level &nbsp;  $($see [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate|"theory section"]]$)$.
 {{ML-Fuß}}

@@ Line 2: / Line 2: @@
 [[File:P_ID2523__KC_A_2_9_neu.png|right|frame|Some Reed-Solomon codes]]
-Adjacent is an incomplete list of possible Reed&ndash;Solomon codes known to be based on a Galois field&nbsp; ${\rm GF}(q) = {\rm GF}(2^m)$&nbsp;. The parameter&nbsp; $m$&nbsp; specifies with how many bits a RS code symbol is represented. It is valid:
+Adjacent is an incomplete list of possible Reed&ndash;Solomon codes known to be based on a Galois field&nbsp; ${\rm GF}(q) = {\rm GF}(2^m)$.&nbsp;
 * $m = 4$&nbsp; (red font),
 * $m = 5$&nbsp; (blue font),
 * $m = 6$&nbsp; (green font).
 A Reed&ndash;Solomon code is generally denoted as follows: &nbsp; ${\rm RSC}\ (n, \ k, \ d_{\rm min})_q$.
 The parameters have the following meaning:
-* $n$&nbsp; specifies the number of symbols of a codeword&nbsp; $\underline{c}$&nbsp; &nbsp; &#8658; &nbsp; <b>length</b> of the code,
+# The parameter&nbsp; $n$&nbsp; specifies the number of symbols of a code word&nbsp; $\underline{c}$&nbsp; &nbsp; &#8658; &nbsp; <b>length</b> of the code.
-* $k$&nbsp; specifies the number of symbols of an information block&nbsp; $\underline{u}$&nbsp; &nbsp; &#8658; &nbsp; <b>dimension</b> of the code,
+# Tthe parameter&nbsp; $k$&nbsp; specifies the number of symbols of an information block&nbsp; $\underline{u}$&nbsp; &nbsp; &#8658; &nbsp; <b>dimension</b>&nbsp; of the code.
-* $d_{\rm min}$&nbsp; denotes the&nbsp; <b> minimum distance </b>&nbsp; between two codewords <br>$($for all Reed&ndash;Solomon codes equal&nbsp; $n-k+1)$,
+# The parameter&nbsp; $d_{\rm min}$&nbsp; denotes the&nbsp; <b> minimum distance </b>&nbsp; between two code words <br>$($for all Reed&ndash;Solomon codes equal&nbsp; $n-k+1)$.
-* $q$&nbsp; gives an indication of the use of the Galois field&nbsp; ${\rm GF}(q)$.
+# The parameter&nbsp; $q$&nbsp; gives an indication of the use of the Galois field&nbsp; ${\rm GF}(q)$.
@@ Line 31: / Line 32: @@
 Hints:
 * This exercise belongs to the chapter&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes| "Definition and Properties of Reed-Solomon Codes"]].
 * However, reference is also made to the chapter&nbsp; [[Channel_Coding/Extension_Field| "Extension Field"]].
@@ Line 38: / Line 40: @@
 ===Solution===
 <quiz display=simple>
-{It is valid&nbsp; $c_i &#8712; {\rm GF}(2^m)$. Which RS code parameters&nbsp; $n$&nbsp; result?
+{It holds&nbsp; $c_i &#8712; {\rm GF}(2^m)$.&nbsp; Which Reed-Solomon code parameters&nbsp; $n$&nbsp; result?
 |type="{}"}
 $m = 4 \text{:} \hspace{0.4cm} n \ = \ ${ 15 }
@@ Line 44: / Line 46: @@
 $m = 6 \text{:} \hspace{0.4cm} n \ = \ ${ 63 }
-{Two specific RS codes&nbsp; ${\rm RSC} \ 1 \ (m = 4, \ t = 4)$&nbsp; and&nbsp; ${\rm RSC} \ 2 \ (m = 5, \ t = 8)$&nbsp; are considered below. <br>Which RS parameter&nbsp; $k$&nbsp; can be used to correct exactly&nbsp; $t$&nbsp; symbol errors?
+{Two specific Reed-Solomon codes &nbsp; ${\rm RSC} \ 1 \ (m = 4, \ t = 4)$ &nbsp; and &nbsp; ${\rm RSC} \ 2 \ (m = 5, \ t = 8)$ &nbsp; are considered below. <br>Which parameter&nbsp; $k$&nbsp; can be used to correct exactly&nbsp; $t$&nbsp; symbol errors?
 |type="{}"}
 ${\rm RSC} \ 1 \text{:} \hspace{0.4cm} k \ = \ ${ 7 }
 ${\rm RSC} \ 2 \text{:} \hspace{0.4cm} k \ = \ ${ 15 }
-{Which designations are correct for&nbsp; $\rm RSC \ 1$&nbsp; and&nbsp; $\rm RSC \ 2$&nbsp; respectively?
+{Which designations are correct for &nbsp; $\rm RSC \ 1$ &nbsp; resp. &nbsp; $\rm RSC \ 2$?
 |type="[]"}
-+ $\rm RSC \ 1$&nbsp; nennt man auch&nbsp; $\rm RSC \, (15, \, 7, \, 9)_{16}$.
++ $\rm RSC \ 1$&nbsp; is also called&nbsp; $\rm RSC \, (15, \, 7, \, 9)_{16}$.
-- $\rm RSC \ 1$&nbsp; nennt man auch&nbsp; $\rm RSC \, (15, \, 7, \, 4)_4$.
+- $\rm RSC \ 1$&nbsp; is also called&nbsp; $\rm RSC \, (15, \, 7, \, 4)_4$.
-- $\rm RSC \ 2$&nbsp; nennt man auch&nbsp; $\rm RSC \, (31, \, 17, \, 15)_{32}$.
+- $\rm RSC \ 2$&nbsp; is also called&nbsp; $\rm RSC \, (31, \, 17, \, 15)_{32}$.
-+ $\rm RSC \ 2$&nbsp; nennt man auch&nbsp; $\rm  RSC \, (31, \, 15, \, 17)_{32}$.
++ $\rm RSC \ 2$&nbsp; is also called&nbsp; $\rm  RSC \, (31, \, 15, \, 17)_{32}$.
 {How many symbol errors&nbsp; $(e)$&nbsp; can be detected at most?
@@ Line 63: / Line 65: @@
 {What are the codes under consideration in binary notation?
 |type="[]"}
-- $\rm RSC \ 1$&nbsp; corresponds to code&nbsp; $\rm RSC \, (60, \, 28, \, 36)_2$.
+- $\rm RSC \ 1$&nbsp; corresponds to the code&nbsp; $\rm RSC \, (60, \, 28, \, 36)_2$.
-+ $\rm RSC \ 1$&nbsp; corresponds to code&nbsp; $\rm RSC \, (60, \, 28, \, 9)_2$.
++ $\rm RSC \ 1$&nbsp; corresponds to the code&nbsp; $\rm RSC \, (60, \, 28, \, 9)_2$.
-+ $\rm RSC \ 2$&nbsp; corresponds to code&nbsp; $\rm RSC \, (155, \, 75, \, 17)_2$.
++ $\rm RSC \ 2$&nbsp; corresponds to the code&nbsp; $\rm RSC \, (155, \, 75, \, 17)_2$.
-- $\rm RSC \ 2$&nbsp; corresponds to code&nbsp; $\rm RSC \, (124, \, 60, \, 17)_2$.
+- $\rm RSC \ 2$&nbsp; corresponds to the code&nbsp; $\rm RSC \, (124, \, 60, \, 17)_2$.
 </quiz>

@@ Line 30: / Line 30: @@
 Hints:
-* Die Aufgabe gehört zum Kapitel&nbsp; [[Channel_Coding/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes| Definition und Eigenschaften von Reed&ndash;Solomon&ndash;Codes]].
+* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes| "Definition and Properties of Reed-Solomon Codes"]].
-* Bezug genommen wird aber auch auf das Kapitel&nbsp; [[Channel_Coding/Erweiterungsk%C3%B6rper| Erweiterungskörper]].
+* However, reference is also made to the chapter&nbsp; [[Channel_Coding/Extension_Field| "Extension Field"]].
-===Fragebogen===
+===Solution===
 <quiz display=simple>
-{Es gelte&nbsp; $c_i &#8712; {\rm GF}(2^m)$. Welche RS&ndash;Codeparameter&nbsp; $n$&nbsp; ergeben sich?
+{It is valid&nbsp; $c_i &#8712; {\rm GF}(2^m)$. Which RS code parameters&nbsp; $n$&nbsp; result?
 |type="{}"}
 $m = 4 \text{:} \hspace{0.4cm} n \ = \ ${ 15 }
@@ Line 44: / Line 44: @@
 $m = 6 \text{:} \hspace{0.4cm} n \ = \ ${ 63 }
-{Im Folgenden werden zwei spezielle RS&ndash;Codes&nbsp; ${\rm RSC} \ 1 \ (m = 4, \ t = 4)$&nbsp; und&nbsp; ${\rm RSC} \ 2 \ (m = 5, \ t = 8)$&nbsp; betrachtet. <br>Mit welchem RS&ndash;Parameter&nbsp; $k$&nbsp; lassen sich genau&nbsp; $t$&nbsp; Symbolfehler korrigeren?
+{Two specific RS codes&nbsp; ${\rm RSC} \ 1 \ (m = 4, \ t = 4)$&nbsp; and&nbsp; ${\rm RSC} \ 2 \ (m = 5, \ t = 8)$&nbsp; are considered below. <br>Which RS parameter&nbsp; $k$&nbsp; can be used to correct exactly&nbsp; $t$&nbsp; symbol errors?
 |type="{}"}
 ${\rm RSC} \ 1 \text{:} \hspace{0.4cm} k \ = \ ${ 7 }
 ${\rm RSC} \ 2 \text{:} \hspace{0.4cm} k \ = \ ${ 15 }
-{Welche Bezeichnungen sind für&nbsp; $\rm RSC \ 1$&nbsp; bzw.&nbsp; $\rm RSC \ 2$&nbsp; richtig?
+{Which designations are correct for&nbsp; $\rm RSC \ 1$&nbsp; and&nbsp; $\rm RSC \ 2$&nbsp; respectively?
 |type="[]"}
 + $\rm RSC \ 1$&nbsp; nennt man auch&nbsp; $\rm RSC \, (15, \, 7, \, 9)_{16}$.
@@ Line 56: / Line 56: @@
 + $\rm RSC \ 2$&nbsp; nennt man auch&nbsp; $\rm  RSC \, (31, \, 15, \, 17)_{32}$.
-{Wieviele Symbolfehler&nbsp; $(e)$&nbsp; können höchstens erkannt werden?
+{How many symbol errors&nbsp; $(e)$&nbsp; can be detected at most?
 |type="{}"}
 ${\rm RSC} \ 1 \text{:} \hspace{0.4cm} e \ = \ ${ 8 }
 ${\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="[]"}
-- $\rm RSC \ 1$&nbsp; entspricht dem Code&nbsp; $\rm RSC \, (60, \, 28, \, 36)_2$.
+- $\rm RSC \ 1$&nbsp; corresponds to code&nbsp; $\rm RSC \, (60, \, 28, \, 36)_2$.
-+ $\rm RSC \ 1$&nbsp; entspricht dem Code&nbsp; $\rm RSC \, (60, \, 28, \, 9)_2$.
++ $\rm RSC \ 1$&nbsp; corresponds to code&nbsp; $\rm RSC \, (60, \, 28, \, 9)_2$.
-+ $\rm RSC \ 2$&nbsp; entspricht dem Code&nbsp; $\rm RSC \, (155, \, 75, \, 17)_2$.
++ $\rm RSC \ 2$&nbsp; corresponds to code&nbsp; $\rm RSC \, (155, \, 75, \, 17)_2$.
-- $\rm RSC \ 2$&nbsp; entspricht dem Code&nbsp; $\rm RSC \, (124, \, 60, \, 17)_2$.
+- $\rm RSC \ 2$&nbsp; corresponds to code&nbsp; $\rm RSC \, (124, \, 60, \, 17)_2$.
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Für die Codelänge der Reed&ndash;Solomon&ndash;Codes gilt allgemein.
+'''(1)'''&nbsp; For the code length of Reed&ndash;Solomon codes, the following applies in general.
 :$$n = q -1 = 2^m -1 \hspace{0.3cm}\Rightarrow  \hspace{0.3cm} m = 4 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 15} \hspace{0.05cm},
 \hspace{0.4cm}m = 5 {\rm :}\hspace{0.2cm} n \hspace{0.15cm}\underline {= 31} \hspace{0.05cm},\hspace{0.4cm}
@@ Line 77: / Line 77: @@
-'''(2)'''&nbsp; Um $t$ Symbolfehler korrigieren zu können, muss die minimale Distanz $d_{\rm min} = 2t + 1$ betragen.
+'''(2)'''&nbsp; To be able to correct $t$ symbol errors, the minimum distance must be $d_{\rm min} = 2t + 1$.
-*Der Reed&ndash;Solomon&ndash;Code ist ein sogenannter MDS&ndash;Code (<i>Maximum Distance Separable</i>).
+*The Reed&ndash;Solomon code is a so-called MDS code (<i>Maximum Distance Separable</i>).
-*Für diese gilt:
+*For this applies:
 :$$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}. $$
-*Damit erhält man für den
+*This gives for the
 :* $\rm RSC \ 1$ (mit $m = 4, \ t = 4) \text{:} \hspace{0.2cm} k = 2^4 - (2 \cdot 4 + 1) \ \underline{= 7}$,
 :* $\rm RSC \ 2$ (mit $m = 5, \ t = 8) \text{:} \hspace{0.2cm} k = 2^5 - (2 \cdot 8 + 1) \ \underline{= 15}$.
@@ Line 88: / Line 88: @@
-'''(3)'''&nbsp; Die Bezeichnung eines Reed&ndash;Solomon&ndash;Codes lautet ${\rm RSC} \, (n, \, k, \, d_{\rm min})_q$ mit $q = 2^m = n + 1$.
+'''(3)'''&nbsp; The denotation of a Reed&ndash;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>:
 * $\rm RSC \ 1 \Rightarrow RSC \, (15, \, 7, \, 9)_{16}$,
 * $\rm RSC \ 2 \Rightarrow RSC \, (31, \, 15, \, 17)_{32}$.
@@ Line 96: / Line 96: @@
-'''(4)'''&nbsp; 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)'''&nbsp; 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:
 * ${\rm RSC} \ 1 \text{:} \hspace{0.2cm} d_{\rm min} = \ \ 9, \ t = 4, \ \underline{e = 8}$,
 * ${\rm RSC} \ 2 \text{:} \hspace{0.2cm} d_{\rm min} = 17, \ t = 8, \ \underline{e = 16}$.
@@ Line 102: / Line 102: @@
-'''(5)'''&nbsp; Richtig sind die beiden mittleren <u>Lösungsvorschläge 2 und 3</u>:
+'''(5)'''&nbsp; 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:
 :* $\rm RSC \ 1 \Rightarrow RSC \, (15, \, 7, \, 9)_{16} \Rightarrow RSC \, (60, \, 28, \, 9)_2$,
 :* $\rm RSC \ 2 \Rightarrow RSC \, (31, \, 15, \, 17)_{32} \Rightarrow RSC \, (155, \, 75, \, 17)_2$.
-*Für die minimale Distanz auf Bitebene &nbsp; &rArr; &nbsp; $\rm GF(2)$ ergeben sich mit $d_{\rm min} = 9$ bzw. $d_{\rm min} = 17$ die gleichen Werte wie auf Symbolebene &nbsp; <br>&rArr; &nbsp; $\rm GF(2^4)$ 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 &nbsp; &rArr; &nbsp; $\rm GF(2)$, $d_{\rm min} = 9$ resp. $d_{\rm min} = 17$ result in the same values as on symbol level &nbsp; <br>&rArr; &nbsp; $\rm GF(2^4)$ resp. $\rm GF(2^5)$ (see [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate|"theory section"]]).
 {{ML-Fuß}}

@@ Line 1: / Line 1: @@
-{{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&ndash;Solomon&ndash;Codes, die bekanntlich auf einem Galoisfeld&nbsp; ${\rm GF}(q) = {\rm GF}(2^m)$&nbsp; basieren. Der Parameter&nbsp; $m$&nbsp; gibt an, mit wie vielen Bit ein RS&ndash;Codesymbol dargestellt wird. Es gilt:
+Adjacent is an incomplete list of possible Reed&ndash;Solomon codes known to be based on a Galois field&nbsp; ${\rm GF}(q) = {\rm GF}(2^m)$&nbsp;. The parameter&nbsp; $m$&nbsp; specifies with how many bits a RS code symbol is represented. It is valid:
-* $m = 4$&nbsp; (rote Schrift),
+* $m = 4$&nbsp; (red font),
-* $m = 5$&nbsp; (blaue Schrift),
+* $m = 5$&nbsp; (blue font),
-* $m = 6$&nbsp; (grüne Schrift).
+* $m = 6$&nbsp; (green font).
-Ein Reed&ndash;Solomon&ndash;Code wird allgemein wie folgt bezeichnet: &nbsp; ${\rm RSC}\ (n, \ k, \ d_{\rm min})_q$.
+A Reed&ndash;Solomon code is generally denoted as follows: &nbsp; ${\rm RSC}\ (n, \ k, \ d_{\rm min})_q$.
-Die Parameter haben folgende Bedeutung:
+The parameters have the following meaning:
-* $n$&nbsp; gibt die Anzahl der Symbole eines Codewortes&nbsp; $\underline{c}$&nbsp; an &nbsp; &#8658; &nbsp; <b>Länge</b> des Codes,
+* $n$&nbsp; specifies the number of symbols of a codeword&nbsp; $\underline{c}$&nbsp; &nbsp; &#8658; &nbsp; <b>length</b> of the code,
-* $k$&nbsp; gibt die Anzahl der Symbole eines Informationsblocks&nbsp; $\underline{u}$&nbsp; an &nbsp; &#8658; &nbsp; <b>Dimension</b> des Codes,
+* $k$&nbsp; specifies the number of symbols of an information block&nbsp; $\underline{u}$&nbsp; &nbsp; &#8658; &nbsp; <b>dimension</b> of the code,
-* $d_{\rm min}$&nbsp; kennzeichnet die&nbsp; <b>minimale Distanz </b>&nbsp; zwischen zwei Codeworten <br>$($bei allen Reed&ndash;Solomon&ndash;Codes gleich&nbsp; $n-k+1)$,
+* $d_{\rm min}$&nbsp; denotes the&nbsp; <b> minimum distance </b>&nbsp; between two codewords <br>$($for all Reed&ndash;Solomon codes equal&nbsp; $n-k+1)$,
-* $q$&nbsp; gibt einen Hinweis auf die Verwendung des Galoisfeldes&nbsp; ${\rm GF}(q)$.
+* $q$&nbsp; gives an indication of the use of the Galois field&nbsp; ${\rm GF}(q)$.
-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&ndash;Codes wird jedes Informations&ndash; und Codesymbol durch&nbsp; $m$&nbsp; Bit dargestellt.
+*In this realization of a RS code, each information and code symbol is represented by&nbsp; $m$&nbsp; bits.
-*Beispielsweise erkennt man aus der ersten Zeile, dass die minimale Distanz hinsichtlich der Bits ebenfalls&nbsp; $d_{\rm min} = 5$&nbsp; ist, wenn in&nbsp; ${\rm GF}(2^m)$&nbsp; die minimale Distanz&nbsp; $d_{\rm min} = 5$&nbsp; beträgt.
+*For example, it can be seen from the first row that the minimum distance in terms of bits is also&nbsp; $d_{\rm min} = 5$&nbsp; if in&nbsp; ${\rm GF}(2^m)$&nbsp; the minimum distance is&nbsp; $d_{\rm min} = 5$&nbsp;.
-*Damit können bis zu&nbsp; $t = 2$&nbsp; Bitfehler (oder Symbolfehler) korrigiert und bis zu&nbsp; $e = 4$&nbsp; Bitfehler (oder Symbolfehler) erkannt werden.
+*This can correct up to&nbsp; $t = 2$&nbsp; bit errors (or symbol errors) and detect up to&nbsp; $e = 4$&nbsp; bit errors (or symbol errors).
@@ Line 29: / Line 29: @@
-''Hinweise:''
+Hints:
 * Die Aufgabe gehört zum Kapitel&nbsp; [[Channel_Coding/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes| Definition und Eigenschaften von Reed&ndash;Solomon&ndash;Codes]].
 * Bezug genommen wird aber auch auf das Kapitel&nbsp; [[Channel_Coding/Erweiterungsk%C3%B6rper| Erweiterungskörper]].

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

← Older revision	Revision as of 12:25, 2 September 2022
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← Older revision		Revision as of 12:48, 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^]]