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

Hwang at 15:28, 23 January 2023

2023-01-23T15:28:47Z

Hwang at 19:03, 1 November 2022

2022-11-01T19:03:22Z

Guenter at 10:43, 12 October 2022

2022-10-12T10:43:27Z

Guenter at 10:03, 12 October 2022

2022-10-12T10:03:40Z

Guenter at 10:02, 12 October 2022

2022-10-12T10:02:53Z

Guenter at 15:40, 11 October 2022

2022-10-11T15:40:52Z

Guenter at 15:22, 4 September 2022

2022-09-04T15:22:39Z

Noah at 18:19, 2 September 2022

2022-09-02T18:19:25Z

Noah at 18:19, 2 September 2022

2022-09-02T18:19:10Z

Guenter at 12:32, 2 September 2022

2022-09-02T12:32:26Z

@@ Line 82: / Line 82: @@
 '''(3)'''&nbsp; From&nbsp; $d_{\rm min} = 33$&nbsp; follows again&nbsp; $t = 16 \ \Rightarrow \ N_{3} \ \underline{= 16}$.
-*If exactly one bit is corrupted in each code symbol,&nbsp; this also means&nbsp; $16$&nbsp; symbol errors.
+*If exactly one bit is falsified in each code symbol,&nbsp; this also means&nbsp; $16$&nbsp; symbol errors.
 *This is the maximum value that the Reed&ndash;Solomon decoder can still handle.
@@ Line 88: / Line 88: @@
-'''(4)'''&nbsp; The Reed&ndash;Solomon decoder can correct&nbsp; $16$&nbsp; corrupted code symbols.
+'''(4)'''&nbsp; The Reed&ndash;Solomon decoder can correct&nbsp; $16$&nbsp; falsified code symbols.
-*It does not matter whether in a code symbol only one bit or all&nbsp; $m = 8$&nbsp; bits have been corrupted.
+*It does not matter whether in a code symbol only one bit or all&nbsp; $m = 8$&nbsp; bits have been falsified.
-*Therefore,&nbsp; with the most favorable error distribution,&nbsp; up to&nbsp; $N_4 = 8 \cdot 16 \ \underline{= 128}$&nbsp; bits can be corrupted without the code word being incorrectly decoded.
+*Therefore,&nbsp; with the most favorable error distribution,&nbsp; up to&nbsp; $N_4 = 8 \cdot 16 \ \underline{= 128}$&nbsp; bits can be falsified without the code word being incorrectly decoded.
 {{ML-Fuß}}

@@ Line 12: / Line 12: @@
 * $k$&nbsp; indicates the&nbsp; "dimension"&nbsp; $($symbol number of an information block$)$,
-* $d_{\rm min}$&nbsp; denotes the&nbsp; "minimum distance"&nbsp; between two codewords.&nbsp;
+* $d_{\rm min}$&nbsp; denotes the&nbsp; "minimum distance"&nbsp; between two code words.&nbsp;
 *For any Reed-Solomon code:

@@ Line 33: / Line 33: @@
 ===Questions===
 <quiz display=simple>
-{Specify the characteristics of the &nbsp; ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_q$&nbsp;.
+{Specify the characteristics of the &nbsp; ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_q$.
 |type="{}"}
 $q \hspace{0.2cm} = \ ${ 256 }
@@ Line 57: / Line 57: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; From the code length $n = 255$ follows $q \ \underline{= 256}$.
+'''(1)'''&nbsp; From the code length&nbsp; $n = 255$&nbsp; follows&nbsp; $q \ \underline{= 256}$.
-*The code rate is given by $R = {223}/{255} \hspace{0.15cm}\underline {=0.8745}\hspace{0.05cm}.$
+*The code rate is given by&nbsp; $R = {223}/{255} \hspace{0.15cm}\underline {=0.8745}\hspace{0.05cm}.$
-*The minimum distance is $d_{\rm min} = n - k +1 = 255 - 223 +1
+*The minimum distance is&nbsp; $d_{\rm min} = n - k +1 = 255 - 223 +1
   \hspace{0.15cm}\underline {=33}\hspace{0.05cm}.$
-*This allows
+*This allows:
-:* $e = d_{\rm min} - 1 \ \underline{= 32}$ symbol errors can be detected, and.
+:* $e = d_{\rm min} - 1 \ \underline{= 32}$&nbsp; symbol errors can be detected, and
-:* $t = e/2$ (rounded down), so $\underline{t = 16}$ symbol errors can be corrected.
-'''(2)'''&nbsp; 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)'''&nbsp; The code&nbsp; $\rm RSC \, (2040, \, 1784, \, d_{\rm min})_2$&nbsp; is the binary representation of the&nbsp; ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_{256}$&nbsp; discussed in&nbsp; '''(1)'''&nbsp;
-'''(3)'''&nbsp; From $d_{\rm min} = 33$ follows again $t = 16 \ \Rightarrow \ N_{3} \ \underline{= 16}$.
+Here&nbsp; $8$&nbsp; bits&nbsp; (1 byte)&nbsp; are used per code symbol.
 *If exactly one bit is corrupted in each code symbol, this also means 16 symbol errors.
 *This is the maximum value that the Reed&ndash;Solomon decoder can still handle.
-'''(4)'''&nbsp; 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)'''&nbsp; The Reed&ndash;Solomon decoder can correct&nbsp; $16$&nbsp; 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&nbsp; $m = 8$&nbsp; bits have been corrupted.
 {{ML-Fuß}}

@@ Line 23: / Line 23: @@
 Hints:
-* The exercise belongs to the chapter&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes| "Definition and properties of Reed&ndash;Solomon Codes"]].
+* The exercise belongs to the chapter&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes| "Definition and Properties of Reed&ndash;Solomon Codes"]].
 * Information relevant to this exercise can be found on the&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes#Code_name_and_code_rate|"Code name and code rate"]] page.

@@ Line 2: / Line 2: @@
 [[File:P_ID2526__KC_Z_2_10.png|right|frame|The inventors of the Reed-Solomon codes]]
-The codes developed by&nbsp; [https://en.wikipedia.org/wiki/Irving_S._Reed "Irving Stoy Reed"]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Gustave_Solomon "Gustave Solomon"]&nbsp; in the early 1960s are referred to in this tutorial as follows:
+The codes developed by&nbsp; [https://en.wikipedia.org/wiki/Irving_S._Reed Irving Stoy Reed]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Gustave_Solomon Gustave Solomon]&nbsp; in the early 1960s are referred to in this tutorial as follows:
 :$${\rm RSC} \, (n, \, k, \, d_{\rm min}) _q.$$
@@ Line 8: / Line 8: @@
 * $q = 2^m$&nbsp; is an indication of the&nbsp; "size"&nbsp; of the Galois field &nbsp; &#8658; &nbsp; ${\rm GF}(q)$,
-* $n = q - 1$&nbsp; is the&nbsp; "code length"&nbsp; (symbol number of a code word),
+* $n = q - 1$&nbsp; is the&nbsp; "code length"&nbsp; $($symbol number of a code word$)$,
-* $k$&nbsp; indicates the&nbsp; "dimension"&nbsp; (symbol number of an information block),
+* $k$&nbsp; indicates the&nbsp; "dimension"&nbsp; $($symbol number of an information block$)$,
 * $d_{\rm min}$&nbsp; denotes the&nbsp; "minimum distance"&nbsp; between two codewords.&nbsp;
@@ Line 24: / Line 24: @@
 Hints:
 * The exercise belongs to the chapter&nbsp; [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes| "Definition and properties of Reed&ndash;Solomon Codes"]].
 * Information relevant to this exercise can be found on the&nbsp; [[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===
 <quiz display=simple>
-{Specify the characteristics of the&nbsp; ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_q$&nbsp;.
+{Specify the characteristics of the &nbsp; ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_q$&nbsp;.
 |type="{}"}
 $q \hspace{0.2cm} = \ ${ 256 }
@@ Line 40: / Line 41: @@
 $d_{\rm min} \ = \ ${ 33 }
-{Specify the characteristics of the&nbsp; $\rm RSC \, (2040, \, 1784, \, d_{\rm min})_2$&nbsp;.
+{Specify the characteristics of the &nbsp; $\rm RSC \, (2040, \, 1784, \, d_{\rm min})_2$&nbsp;.
 |type="{}"}
 $R \hspace{0.2cm} = \ ${ 0.8745 3% }
 $d_{\rm min} \ = \ ${ 33 }
-{How many bit errors&nbsp; $(N_3)$&nbsp; may a received word&nbsp; $\underline{y}$&nbsp; have at most, so that it is <u>certainly decoded correctly</u>?
+{How many bit errors&nbsp; $(N_3)$&nbsp; may a received word&nbsp; $\underline{y}$&nbsp; have at most,&nbsp; so that it is&nbsp; <u>certainly decoded correctly</u>?
 |type="{}"}
 $N_{3} \ = \ $ { 16 }
-{How many bit errors&nbsp; $(N_4)$&nbsp; may a received word&nbsp; $\underline{y}$&nbsp; have in the best case so that it could still be <u>correctly decoded</u>?
+{How many bit errors&nbsp; $(N_4)$&nbsp; may a received word&nbsp; $\underline{y}$&nbsp; have&nbsp; <u>in the best case</u>&nbsp; so that it could still be&nbsp; <u>correctly decoded</u>?
 |type="{}"}
 $N_{4} \ = \ $ { 128 }

← Older revision		Revision as of 15:40, 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'''.

@@ 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_ID2526__KC_Z_2_10.png|right|frame|Die beiden Erfinder der Reed–Solomon–Codes]]
+[[File:P_ID2526__KC_Z_2_10.png|right|frame|The two inventors of the Reed-Solomon codes]]
-Die von&nbsp; [https://de.wikipedia.org/wiki/Irving_Stoy_Reed Irving Stoy Reed]&nbsp; und&nbsp; [https://de.wikipedia.org/wiki/Gustave_Solomon Gustave Solomon]&nbsp; Anfang der 1960er Jahre entwickelten Codes werden in diesem Tutorial wie folgt bezeichnet:
+The codes developed by&nbsp; [https://en.wikipedia.org/wiki/Irving_S._Reed "Irving Stoy Reed"]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Gustave_Solomon "Gustave Solomon"]&nbsp; in the early 1960s are referred to in this tutorial as follows:
 :$${\rm RSC} \, (n, \, k, \, d_{\rm min}) _q.$$
-Die Codeparameter haben folgende Bedeutungen:
+The code parameters have the following meanings:
-* $q = 2^m$&nbsp; ist ein Hinweis auf die Größe des Galoisfeldes &nbsp; &#8658; &nbsp; ${\rm GF}(q)$,
+* $q = 2^m$&nbsp; is an indication of the size of the Galois field &nbsp; &#8658; &nbsp; ${\rm GF}(q)$,
-* $n = q - 1$&nbsp; ist die Codelänge (Symbolanzahl eines Codewortes),
+* $n = q - 1$&nbsp; is the code length (symbol number of a code word),
-* $k$&nbsp; gibt die Dimension an (Symbolanzahl eines Informationsblocks),
+* $k$&nbsp; indicates the dimension (symbol number of an information block),
-* $d_{\rm min}$&nbsp; bezeichnet die minimale Distanz zwischen zwei Codeworten. Für jeden Reed–Solomon–Codes gilt&nbsp; $d_{\rm min} = n - k + 1$.
+* $d_{\rm min}$&nbsp; denotes the minimum distance between two codewords. For any Reed-Solomon code, $d_{\rm min} = n - k + 1$.
-*Mit keinem anderen Code mit gleichem&nbsp; $k$&nbsp; und&nbsp;  $n$&nbsp; ergibt sich ein größerer Wert.
+*No other code with the same&nbsp; $k$&nbsp; and&nbsp; $n$&nbsp; yields a larger value.
@@ Line 19: / Line 19: @@
+===Questions===
 <quiz display=simple>
-{Geben Sie die Kenngrößen des&nbsp; ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_q$&nbsp; an.
+{Specify the characteristics of the&nbsp; ${\rm RSC} \, (255, \, 223, \, d_{\rm min})_q$&nbsp;.
 |type="{}"}
 $q \hspace{0.2cm} = \ ${ 256 }
@@ Line 38: / Line 37: @@
 $d_{\rm min} \ = \ ${ 33 }
-{Geben Sie die Kenngrößen des&nbsp; $\rm RSC \, (2040, \, 1784, \, d_{\rm min})_2$&nbsp; an.
+{Specify the characteristics of the&nbsp; $\rm RSC \, (2040, \, 1784, \, d_{\rm min})_2$&nbsp;.
 |type="{}"}
 $R \hspace{0.2cm} = \ ${ 0.8745 3% }
 $d_{\rm min} \ = \ ${ 33 }
-{Wieviele Bitfehler&nbsp; $(N_3)$&nbsp; darf ein Empfangswort&nbsp; $\underline{y}$&nbsp; maximal aufweisen, damit es <u>mit Sicherheit richtig decodiert wird</u>?
+{How many bit errors&nbsp; $(N_3)$&nbsp; may a received word&nbsp; $\underline{y}$&nbsp; have at most, so that it is <u>certainly decoded correctly</u>?
 |type="{}"}
 $N_{3} \ = \ $ { 16 }
-{Wieviele Bitfehler&nbsp; $(N_4)$&nbsp; darf ein Empfangswort&nbsp; $\underline{y}$&nbsp; im günstigsten Fall aufweisen, damit es noch <u>richtig decodiert werden könnte</u>?
+{How many bit errors&nbsp; $(N_4)$&nbsp; may a received word&nbsp; $\underline{y}$&nbsp; have in the best case so that it could still be <u>correctly decoded</u>?
 |type="{}"}
 $N_{4} \ = \ $ { 128 }
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Aus der Codelänge $n = 255$ folgt $q \ \underline{= 256}$.
+'''(1)'''&nbsp; From the code length $n = 255$ follows $q \ \underline{= 256}$.
-*Die Coderate ergibt sich zu $R = {223}/{255} \hspace{0.15cm}\underline {=0.8745}\hspace{0.05cm}.$
+*The code rate is given by $R = {223}/{255} \hspace{0.15cm}\underline {=0.8745}\hspace{0.05cm}.$
-*Die minimale Distanz beträgt $d_{\rm min} = n - k +1 = 255 - 223 +1
+*The minimum distance is $d_{\rm min} = n - k +1 = 255 - 223 +1
   \hspace{0.15cm}\underline {=33}\hspace{0.05cm}.$
-*Damit können
+*This allows
-:* $e = d_{\rm min} - 1 \ \underline{= 32}$ Symbolfehler erkannt werden, und
+:* $e = d_{\rm min} - 1 \ \underline{= 32}$ symbol errors can be detected, and.
-:* $t = e/2$ (abgerundet), also $\underline{t = 16}$ Symbolfehler korrigiert werden.
+:* $t = e/2$ (rounded down), so $\underline{t = 16}$ symbol errors can be corrected.
-'''(4)'''&nbsp; Der RS&ndash;Decoder kann 16 verfälschte Codesymbole korrigieren,
+'''(4)'''&nbsp; The RS decoder can correct 16 corrupted code symbols,
-*wobei es egal ist, ob in einem Codesymbol nur ein Bit oder alle $m = 8$ Bit verfälscht wurden.
+*whereby it does not matter whether in a code symbol only one bit or all $m = 8$ bits have been corrupted.
-*Deshalb können bei der günstigsten Fehlerverteilung bis zu $N_4 = 8 \cdot 16 \ \underline{= 128}$ Bit verfälscht sein, ohne dass das Codewort falsch decodiert wird.
+*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}}

← Older revision		Revision as of 12:32, 2 September 2022
Line 84:		Line 84:


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