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Kanalcodierung/Definition und Eigenschaften von Reed–Solomon–Codes - Versionsgeschichte
2024-03-28T13:19:20Z
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Guenter am 7. Oktober 2022 um 16:11 Uhr
2022-10-07T16:11:13Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 7. Oktober 2022, 16:11 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l372" >Zeile 372:</td>
<td colspan="2" class="diff-lineno">Zeile 372:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Ein weiterer Entwurfsparameter ist die Coderate&nbsp; $R=k/n$,&nbsp; wobei die Codewortlänge&nbsp; $n = 2^m -1$&nbsp; nicht völlig frei wählbar ist.&nbsp; Durch Erweiterung,&nbsp; Verkürzung und Punktierung &ndash; siehe&nbsp; [[Aufgaben:Aufgabe_1.09Z:_Erweiterung_und/oder_Punktierung|"Aufgabe 1.9Z"]]&nbsp; &ndash; kann die Vielzahl an möglichen Codes weiter vergrößert werden.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Ein weiterer Entwurfsparameter ist die Coderate&nbsp; $R=k/n$,&nbsp; wobei die Codewortlänge&nbsp; $n = 2^m -1$&nbsp; nicht völlig frei wählbar ist.&nbsp; Durch Erweiterung,&nbsp; Verkürzung und Punktierung &ndash; siehe&nbsp; [[Aufgaben:Aufgabe_1.09Z:_Erweiterung_und/oder_Punktierung|"Aufgabe 1.9Z"]]&nbsp; &ndash; kann die Vielzahl an möglichen Codes weiter vergrößert werden.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Bei Reed&ndash;Solomon&ndash;Codes ist die Gewichtsverteilung exakt bekannt und es ist eine Anpassung an die Fehlerstruktur des Kanals möglich.&nbsp; Diese Codes sind insbesondere für Bündelfehlerkanäle gut geeignet,&nbsp; die bei mobilen Funksystemen aufgrund temporärer Abschattungen häufig vorliegen.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Bei Reed&ndash;Solomon&ndash;Codes ist die Gewichtsverteilung exakt bekannt und es ist eine Anpassung an die Fehlerstruktur des Kanals möglich.&nbsp; Diese Codes sind insbesondere für Bündelfehlerkanäle gut geeignet,&nbsp; die bei mobilen Funksystemen aufgrund temporärer Abschattungen häufig vorliegen.<br></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div># Im Falle statistisch unabhängiger Fehler sind so genannte &nbsp; [https://de.wikipedia.org/wiki/BCH-Code BCH&ndash;Codes]&nbsp; $($von $\rm B$ose&ndash;$\rm C$haudhuri&ndash;$\rm H$ocquenghem$)$&nbsp; besser geeignet.&nbsp; Diese sind mit den Reed&ndash;Solomon&ndash;Codes eng verwandt,&nbsp; allerdings erfüllen sie nicht immer das Singleton&ndash;Kriterium.&nbsp; Eine ausführliche Beschreibung finden Sie in&nbsp; [Fri96]<ref name='Fri96'>Friedrichs, B.:&nbsp; Kanalcodierung – Grundlagen und Anwendungen in modernen Kommunikations- systemen.&nbsp; Berlin – Heidelberg: Springer, 1996.</ref>.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div># Im Falle statistisch unabhängiger Fehler sind so genannte &nbsp; [https://de.wikipedia.org/wiki/BCH-Code <ins class="diffchange diffchange-inline">"</ins>BCH&ndash;Codes<ins class="diffchange diffchange-inline">"</ins>]&nbsp; $($von $\rm B$ose&ndash;$\rm C$haudhuri&ndash;$\rm H$ocquenghem$)$&nbsp; besser geeignet.&nbsp; Diese sind mit den Reed&ndash;Solomon&ndash;Codes eng verwandt,&nbsp; allerdings erfüllen sie nicht immer das Singleton&ndash;Kriterium.&nbsp; Eine ausführliche Beschreibung finden Sie in&nbsp; [Fri96]<ref name='Fri96'>Friedrichs, B.:&nbsp; Kanalcodierung – Grundlagen und Anwendungen in modernen Kommunikations- systemen.&nbsp; Berlin – Heidelberg: Springer, 1996.</ref>.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Die Decodierung nach dem&nbsp; [https://en.wikipedia.org/wiki/Lattice_problem "BDD&ndash;Prinzip"]&nbsp; ("Bounded Distance Decoding")&nbsp; kann rechentechnisch sehr einfach erfolgen,&nbsp; zum Beispiel mit dem&nbsp; [https://de.wikipedia.org/wiki/Berlekamp-Massey-Algorithmus "Berlekamp&ndash;Massey&ndash;Algorithmus"].&nbsp; Zudem kann der Decoder ohne großen Mehraufwand auch&nbsp; [https://en.wikipedia.org/wiki/Soft-decision_decoder "Soft Decision Information"]&nbsp; verarbeiten.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Die Decodierung nach dem&nbsp; [https://en.wikipedia.org/wiki/Lattice_problem "BDD&ndash;Prinzip"]&nbsp; ("Bounded Distance Decoding")&nbsp; kann rechentechnisch sehr einfach erfolgen,&nbsp; zum Beispiel mit dem&nbsp; [https://de.wikipedia.org/wiki/Berlekamp-Massey-Algorithmus "Berlekamp&ndash;Massey&ndash;Algorithmus"].&nbsp; Zudem kann der Decoder ohne großen Mehraufwand auch&nbsp; [https://en.wikipedia.org/wiki/Soft-decision_decoder "Soft Decision Information"]&nbsp; verarbeiten.<br></div></td></tr>
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Guenter
https://www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33627&oldid=prev
Guenter am 7. Oktober 2022 um 16:09 Uhr
2022-10-07T16:09:03Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 7. Oktober 2022, 16:09 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l283" >Zeile 283:</td>
<td colspan="2" class="diff-lineno">Zeile 283:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P ID2520 KC T 2 3 S3 v1.png|right|frame|Zur Herleitung des Distanzspektrums für den&nbsp; $\text{RSC (3, 2, 2)}_4$|class=fit]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P ID2520 KC T 2 3 S3 v1.png|right|frame|Zur Herleitung des Distanzspektrums für den&nbsp; $\text{RSC (3, 2, 2)}_4$|class=fit]]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&rArr; &nbsp; Aus der oberen Tabelle kann unter anderem abgelesen werden:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&rArr; &nbsp; Aus der oberen Tabelle kann unter anderem abgelesen werden:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l289" >Zeile 289:</td>
<td colspan="2" class="diff-lineno">Zeile 288:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$W_2 = 9,\ \ W_3 = 6.$$ </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$W_2 = 9,\ \ W_3 = 6.$$ </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Es gibt kein einziges Codewort mit nur einer Null. Das heißt: &nbsp; Die minimale Distanz beträgt hier&nbsp; </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Es gibt kein einziges Codewort mit nur einer Null. Das heißt: &nbsp; Die minimale Distanz beträgt hier&nbsp; </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$d_{\rm min} = 2.$$ <del class="diffchange diffchange-inline"><br></del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$d_{\rm min} = 2.$$ </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&rArr; &nbsp; Aus der unteren Tabelle erkennt man, dass auch für die Binärdarstellung&nbsp; $d_{\rm min} = 2$&nbsp; gilt.}}<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&rArr; &nbsp; Aus der unteren Tabelle erkennt man, dass auch für die Binärdarstellung&nbsp; $d_{\rm min} = 2$&nbsp; gilt.}}<br></div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33626&oldid=prev
Guenter am 7. Oktober 2022 um 16:07 Uhr
2022-10-07T16:07:02Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 7. Oktober 2022, 16:07 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l280" >Zeile 280:</td>
<td colspan="2" class="diff-lineno">Zeile 280:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Beispiel 5:}$&nbsp; Die Tabelle verdeutlicht die Bestimmung des Distanzspektrums für den&nbsp; $\text{RSC (3, 2, 2)}_4$. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Beispiel 5:}$&nbsp; Die Tabelle verdeutlicht die Bestimmung des Distanzspektrums für den&nbsp; $\text{RSC (3, 2, 2)}_4$. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Gegenüber den bisherigen Grafiken wird die Symbolmenge vereinfachend mit&nbsp; $\{0,\ 1,\ 2,\ 3\}$&nbsp; anstelle von $\{0,\ \alpha^0,\ \alpha^1,\ \alpha^2\}$&nbsp; bezeichnet. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Gegenüber den bisherigen Grafiken wird die Symbolmenge vereinfachend mit&nbsp; $\{0,\ 1,\ 2,\ 3\}$&nbsp; anstelle von $\{0,\ \alpha^0,\ \alpha^1,\ \alpha^2\}$&nbsp; bezeichnet. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Die Distanz&nbsp; $d$&nbsp; zwischen&nbsp; $\underline {c}_j$&nbsp; und dem Nullwort&nbsp; $\underline {c}_0$&nbsp; ist identisch dem&nbsp; [[Kanalcodierung/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung|Hamming&ndash;Gewicht]]&nbsp; $w_{\rm H}(\underline {c}_j)$.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Die Distanz&nbsp; $d$&nbsp; zwischen&nbsp; $\underline {c}_j$&nbsp; und dem Nullwort&nbsp; $\underline {c}_0$&nbsp; ist identisch dem&nbsp; [[Kanalcodierung/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung|<ins class="diffchange diffchange-inline">"</ins>Hamming&ndash;Gewicht<ins class="diffchange diffchange-inline">"</ins>]]&nbsp; $w_{\rm H}(\underline {c}_j)$.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P ID2520 KC T 2 3 S3 v1.png|right|frame|Zur Herleitung des Distanzspektrums für den&nbsp; $\text{RSC (3, 2, 2)}_4$|class=fit]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P ID2520 KC T 2 3 S3 v1.png|right|frame|Zur Herleitung des Distanzspektrums für den&nbsp; $\text{RSC (3, 2, 2)}_4$|class=fit]]</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l301" >Zeile 301:</td>
<td colspan="2" class="diff-lineno">Zeile 301:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Bei&nbsp; "fehlerkorrigierenden Codes"&nbsp; wählt man meist&nbsp; $d_{\rm min} $&nbsp; ungeradzahlig &nbsp; &#8658; &nbsp; $n-k$&nbsp; geradzahlig.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Bei&nbsp; "fehlerkorrigierenden Codes"&nbsp; wählt man meist&nbsp; $d_{\rm min} $&nbsp; ungeradzahlig &nbsp; &#8658; &nbsp; $n-k$&nbsp; geradzahlig.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Bei Reed&ndash;Solomon&ndash;Codes können dann bis zu &nbsp; $t =(n-k)/2$&nbsp; Symbolfehler korrigiert werden.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Bei Reed&ndash;Solomon&ndash;Codes können dann bis zu &nbsp; $t =(n-k)/2$&nbsp; Symbolfehler korrigiert werden.<br></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div># Die&nbsp; [https://de.wikipedia.org/wiki/Singleton-Schranke Singleton&ndash;Schranke]&nbsp; besagt,&nbsp; dass für alle linearen Codes&nbsp; $d_{\rm min} \le n-k+1$&nbsp; gilt. </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div># Die&nbsp; [https://de.wikipedia.org/wiki/Singleton-Schranke <ins class="diffchange diffchange-inline">"</ins>Singleton&ndash;Schranke<ins class="diffchange diffchange-inline">"</ins>]&nbsp; besagt,&nbsp; dass für alle linearen Codes&nbsp; $d_{\rm min} \le n-k+1$&nbsp; gilt. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div># Reed&ndash;Solomon&ndash;Codes erreichen diese Schranke mit Gleichheit;&nbsp; sie sind so genannte&nbsp; [https://de.wikipedia.org/wiki/MDS-Code MDS&ndash;Codes]&nbsp; ("Maximum Distance Separable").</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div># Reed&ndash;Solomon&ndash;Codes erreichen diese Schranke mit Gleichheit;&nbsp; sie sind so genannte&nbsp; [https://de.wikipedia.org/wiki/MDS-Code <ins class="diffchange diffchange-inline">"</ins>MDS&ndash;Codes<ins class="diffchange diffchange-inline">"</ins>]&nbsp; ("Maximum Distance Separable").</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div># Das&nbsp; [[Kanalcodierung/Schranken_f%C3%BCr_die_Blockfehlerwahrscheinlichkeit#Distanzspektrum_eines_linearen_Codes| Distanzspektrum]]&nbsp; setzt sich zusammen aus&nbsp; $W_0 = 1$&nbsp; sowie weiteren Gewichtsfaktoren&nbsp; $W_i$&nbsp; mit&nbsp; $d &#8804; i &#8804; n$,&nbsp; wobei in der folgenden Gleichung&nbsp; $d_{\rm min}$&nbsp; mit&nbsp; $d$&nbsp; abgekürzt ist:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div># Das&nbsp; [[Kanalcodierung/Schranken_f%C3%BCr_die_Blockfehlerwahrscheinlichkeit#Distanzspektrum_eines_linearen_Codes| <ins class="diffchange diffchange-inline">"</ins>Distanzspektrum<ins class="diffchange diffchange-inline">"</ins>]]&nbsp; setzt sich zusammen aus&nbsp; $W_0 = 1$&nbsp; sowie weiteren Gewichtsfaktoren&nbsp; $W_i$&nbsp; mit&nbsp; $d &#8804; i &#8804; n$,&nbsp; wobei in der folgenden Gleichung&nbsp; $d_{\rm min}$&nbsp; mit&nbsp; $d$&nbsp; abgekürzt ist:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:::<math>W_i = {n \choose i} \cdot \sum_{j = 0}^{i-d}\hspace{0.15cm}(-1)^j \cdot {i \choose j} \cdot \bigg [\hspace{0.03cm}q^{i\hspace{0.03cm}-\hspace{0.03cm}j\hspace{0.03cm}-\hspace{0.03cm}d\hspace{0.03cm}+\hspace{0.03cm}1}-1 \hspace{0.03cm} \bigg ]\hspace{0.05cm}.</math>}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:::<math>W_i = {n \choose i} \cdot \sum_{j = 0}^{i-d}\hspace{0.15cm}(-1)^j \cdot {i \choose j} \cdot \bigg [\hspace{0.03cm}q^{i\hspace{0.03cm}-\hspace{0.03cm}j\hspace{0.03cm}-\hspace{0.03cm}d\hspace{0.03cm}+\hspace{0.03cm}1}-1 \hspace{0.03cm} \bigg ]\hspace{0.05cm}.</math>}}</div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33625&oldid=prev
Guenter am 7. Oktober 2022 um 16:04 Uhr
2022-10-07T16:04:08Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 7. Oktober 2022, 16:04 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l271" >Zeile 271:</td>
<td colspan="2" class="diff-lineno">Zeile 271:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Singleton–Schranke und minimale Distanz ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Singleton–Schranke und minimale Distanz ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Eine wichtige Kenngröße eines jeden Blockcodes ist die&nbsp; [[Kanalcodierung/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung| minimale Distanz]]&nbsp; zwischen zwei beliebigen Codeworten&nbsp; $\underline {c}_i$&nbsp; und&nbsp; $\underline {c}_j$. </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Eine wichtige Kenngröße eines jeden Blockcodes ist die&nbsp; [[Kanalcodierung/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung| <ins class="diffchange diffchange-inline">"</ins>minimale Distanz<ins class="diffchange diffchange-inline">"</ins>]]&nbsp; zwischen zwei beliebigen Codeworten&nbsp; $\underline {c}_i$&nbsp; und&nbsp; $\underline {c}_j$. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Reed&ndash;Solomon&ndash;Codes gehören zur Klasse der&nbsp; '''linearen</i> und&nbsp; zyklischen'''&nbsp; Codes.&nbsp; Bei diesen kann man vom Nullwort&nbsp; $\underline {c}_0 = (0, 0, \hspace{0.05cm} \text{...} \hspace{0.05cm}, 0)$&nbsp; als Bezugsgröße ausgehen. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Reed&ndash;Solomon&ndash;Codes gehören zur Klasse der&nbsp; '''linearen</i> und&nbsp; zyklischen'''&nbsp; Codes.&nbsp; Bei diesen kann man vom Nullwort&nbsp; $\underline {c}_0 = (0, 0, \hspace{0.05cm} \text{...} \hspace{0.05cm}, 0)$&nbsp; als Bezugsgröße ausgehen. </div></td></tr>
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</table>
Guenter
https://www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33624&oldid=prev
Guenter am 7. Oktober 2022 um 16:02 Uhr
2022-10-07T16:02:56Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 7. Oktober 2022, 16:02 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l199" >Zeile 199:</td>
<td colspan="2" class="diff-lineno">Zeile 199:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT=</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT=</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&rArr; &nbsp; Diee&nbsp;[[Kanalcodierung/Allgemeine_Beschreibung_linearer_Blockcodes#Codefestlegung_durch_die_Generatormatrix| "Generatormatrix"]]&nbsp; $\boldsymbol{\rm G}$&nbsp; $(k$ Zeilen,&nbsp; $n$ Spalten$)$&nbsp; und&nbsp; [[Kanalcodierung/Allgemeine_Beschreibung_linearer_Blockcodes#Codefestlegung_durch_die_Pr.C3.BCfmatrix| Prüfmatrix]]&nbsp; $\boldsymbol{\rm H}$&nbsp; $(n-k$ Zeilen,&nbsp; $n$&nbsp; Spalten$)$ müssen gemeinsam folgende Gleichung erfüllen:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&rArr; &nbsp; Diee&nbsp;[[Kanalcodierung/Allgemeine_Beschreibung_linearer_Blockcodes#Codefestlegung_durch_die_Generatormatrix| "Generatormatrix"]]&nbsp; $\boldsymbol{\rm G}$&nbsp; $(k$ Zeilen,&nbsp; $n$ Spalten$)$&nbsp; und&nbsp; [[Kanalcodierung/Allgemeine_Beschreibung_linearer_Blockcodes#Codefestlegung_durch_die_Pr.C3.BCfmatrix| <ins class="diffchange diffchange-inline">"</ins>Prüfmatrix<ins class="diffchange diffchange-inline">"</ins>]]&nbsp; $\boldsymbol{\rm H}$&nbsp; $(n-k$ Zeilen,&nbsp; $n$&nbsp; Spalten$)$ müssen gemeinsam folgende Gleichung erfüllen:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$${\boldsymbol{\rm G} } \cdot { \boldsymbol{\rm H } }^{\rm T}= { \boldsymbol{\rm 0} }\hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$${\boldsymbol{\rm G} } \cdot { \boldsymbol{\rm H } }^{\rm T}= { \boldsymbol{\rm 0} }\hspace{0.05cm}.$$</div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33623&oldid=prev
Guenter am 7. Oktober 2022 um 16:00 Uhr
2022-10-07T16:00:18Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 7. Oktober 2022, 16:00 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l56" >Zeile 56:</td>
<td colspan="2" class="diff-lineno">Zeile 56:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Definition:}$&nbsp; Ein&nbsp; $(n,\ k)$'''&ndash;Reed&ndash;Solomon&ndash;Code'''&nbsp; für das Galoisfeld&nbsp; ${\rm GF}(2^m)$&nbsp; wird festgelegt durch</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Definition:}$&nbsp; Ein&nbsp; $(n,\ k)$'''&ndash;Reed&ndash;Solomon&ndash;Code'''&nbsp; für das Galoisfeld&nbsp; ${\rm GF}(2^m)$&nbsp; wird festgelegt durch</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*die&nbsp; $n= 2^{m-1}$&nbsp; Elemente von&nbsp; ${\rm GF}(2^m) \hspace{-0.05cm}\setminus \hspace{-0.05cm} \{0\} =\{ \alpha^0, \alpha^1, \, \text{...} \, , \alpha^{n-1}\}$,&nbsp; wobei&nbsp; $\alpha$&nbsp; ein&nbsp; [[Kanalcodierung/Erweiterungskörper#Bin.C3.A4re_Erweiterungsk.C3.B6rper_.E2.80.93_Primitive_Polynome|primitives Element]]&nbsp; von&nbsp; ${\rm GF}(2^m)$&nbsp; bezeichnet,</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*die&nbsp; $n= 2^{m-1}$&nbsp; Elemente von&nbsp; ${\rm GF}(2^m) \hspace{-0.05cm}\setminus \hspace{-0.05cm} \{0\} =\{ \alpha^0, \alpha^1, \, \text{...} \, , \alpha^{n-1}\}$,&nbsp; wobei&nbsp; $\alpha$&nbsp; ein&nbsp; [[Kanalcodierung/Erweiterungskörper#Bin.C3.A4re_Erweiterungsk.C3.B6rper_.E2.80.93_Primitive_Polynome|<ins class="diffchange diffchange-inline">"</ins>primitives Element<ins class="diffchange diffchange-inline">"</ins>]]&nbsp; von&nbsp; ${\rm GF}(2^m)$&nbsp; bezeichnet,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*ein an den Informationsblockt&nbsp; $\underline{u}$&nbsp; angepasstes&nbsp; [[Kanalcodierung/Erweiterungskörper#Verallgemeinerte_Definition_eines_Erweiterungsk.C3.B6rpers|Polynom]]&nbsp; vom Grad&nbsp; $k-1$&nbsp; der Form</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*ein an den Informationsblockt&nbsp; $\underline{u}$&nbsp; angepasstes&nbsp; [[Kanalcodierung/Erweiterungskörper#Verallgemeinerte_Definition_eines_Erweiterungsk.C3.B6rpers|<ins class="diffchange diffchange-inline">"</ins>Polynom<ins class="diffchange diffchange-inline">"</ins>]]&nbsp; vom Grad&nbsp; $k-1$&nbsp; der Form</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>u(x) = \sum_{i = 0}^{k-1} u_i \cdot x^{i} = u_0 + u_1 \cdot x + u_2 \cdot x^2 + \hspace{0.1cm}... \hspace{0.1cm}+ u_{k-1} \cdot x^{k-1} \hspace{0.05cm},\hspace{0.4cm} u_i \in {\rm GF}(2^m)</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>u(x) = \sum_{i = 0}^{k-1} u_i \cdot x^{i} = u_0 + u_1 \cdot x + u_2 \cdot x^2 + \hspace{0.1cm}... \hspace{0.1cm}+ u_{k-1} \cdot x^{k-1} \hspace{0.05cm},\hspace{0.4cm} u_i \in {\rm GF}(2^m)</div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33622&oldid=prev
Guenter am 7. Oktober 2022 um 15:47 Uhr
2022-10-07T15:47:47Z
<p></p>
<a href="//www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33622&oldid=33621">Änderungen zeigen</a>
Guenter
https://www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33621&oldid=prev
Guenter am 7. Oktober 2022 um 12:21 Uhr
2022-10-07T12:21:06Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 7. Oktober 2022, 12:21 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l198" >Zeile 198:</td>
<td colspan="2" class="diff-lineno">Zeile 198:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*der Codewortlänge&nbsp; $n$&nbsp; (Symbolanzahl pro Codewort).<br><br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*der Codewortlänge&nbsp; $n$&nbsp; (Symbolanzahl pro Codewort).<br><br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Kanalcodierung/Allgemeine_Beschreibung_linearer_Blockcodes#Codefestlegung_durch_die_Generatormatrix| Generatormatrix]]&nbsp; $\boldsymbol{\rm G}$&nbsp; $(k$ Zeilen,&nbsp; $n$ Spalten$)$&nbsp; und&nbsp; [[Kanalcodierung/Allgemeine_Beschreibung_linearer_Blockcodes#Codefestlegung_durch_die_Pr.C3.BCfmatrix| Prüfmatrix]]&nbsp; $\boldsymbol{\rm H}$&nbsp; $(n-k$ Zeilen,&nbsp; $n$&nbsp; Spalten$)$ müssen gemeinsam folgende Gleichung erfüllen:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{{BlaueBox|TEXT=</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">&rArr; &nbsp; Diee&nbsp;</ins>[[Kanalcodierung/Allgemeine_Beschreibung_linearer_Blockcodes#Codefestlegung_durch_die_Generatormatrix| <ins class="diffchange diffchange-inline">"</ins>Generatormatrix<ins class="diffchange diffchange-inline">"</ins>]]&nbsp; $\boldsymbol{\rm G}$&nbsp; $(k$ Zeilen,&nbsp; $n$ Spalten$)$&nbsp; und&nbsp; [[Kanalcodierung/Allgemeine_Beschreibung_linearer_Blockcodes#Codefestlegung_durch_die_Pr.C3.BCfmatrix| Prüfmatrix]]&nbsp; $\boldsymbol{\rm H}$&nbsp; $(n-k$ Zeilen,&nbsp; $n$&nbsp; Spalten$)$ müssen gemeinsam folgende Gleichung erfüllen:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<del class="diffchange diffchange-inline">:<math></del>{ \boldsymbol{\rm G}} \cdot { \boldsymbol{\rm H }}^{\rm T}= { \boldsymbol{\rm 0}}\hspace{0.05cm}.<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<ins class="diffchange diffchange-inline">$$</ins>{\boldsymbol{\rm G} } \cdot { \boldsymbol{\rm H } }^{\rm T}= { \boldsymbol{\rm 0} }\hspace{0.05cm}.<ins class="diffchange diffchange-inline">$$</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Hierbei bezeichnet&nbsp; $\boldsymbol{\rm 0}$&nbsp; eine Nullmatrix $($alle Elemente gleich $0)$&nbsp; mit&nbsp; $k$&nbsp; Zeilen und&nbsp; $n-k$&nbsp; Spalten.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Hierbei bezeichnet&nbsp; <ins class="diffchange diffchange-inline">"</ins>$\boldsymbol{\rm 0}$<ins class="diffchange diffchange-inline">"</ins>&nbsp; eine Nullmatrix<ins class="diffchange diffchange-inline">:&nbsp; </ins>$($alle Elemente gleich $0)$&nbsp; mit&nbsp; $k$&nbsp; Zeilen und&nbsp; $n-k$&nbsp; Spalten.<ins class="diffchange diffchange-inline">}}</ins><br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{GraueBox|TEXT= </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{GraueBox|TEXT= </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>$\text{Beispiel 4:}$&nbsp; Wir betrachten den&nbsp; $\text{RSC (7, 3, 5)}_8$ <del class="diffchange diffchange-inline">&nbsp; &#8658; &nbsp; Codeparameter</del>&nbsp; $n= 7$,&nbsp; $k= 3$, <del class="diffchange diffchange-inline">basierend auf dem Galoisfeld&nbsp; $\rm GF(2^3 = 8)$&nbsp; mit der Nebenbedingung&nbsp; $\alpha^3 =\alpha + 1$. Beachten Sie hinsichtlich der Codebezeichnung:</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\text{Beispiel 4:}$&nbsp; Wir betrachten den&nbsp; $\text{RSC (7, 3, 5)}_8$<ins class="diffchange diffchange-inline">.</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*Der dritte Parameter der für Blockcodes üblichen Nomenklatur nennt die freie Distanz&nbsp; $d_{\rm min}= 5$.<br></del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*odeparameter</ins>&nbsp; $n= 7$,&nbsp; $k= 3$, </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Anders als bei den im Kapitel&nbsp; [[Kanalcodierung/Beispiele_binärer_Blockcodes|Beispiele binärer Blockcodes]]&nbsp; behandelten binären Codes (Single Parity–check Code, Repetition Code, Hamming Code) wird bei den Reed&ndash;Solomon&ndash;Codes noch der Hinweis&nbsp; $q$&nbsp; zum Galoisfeld hinzugefügt&nbsp; $($hier: &nbsp; $q = 8)$.<br><br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*<ins class="diffchange diffchange-inline">basierend auf dem Galoisfeld&nbsp; $\rm GF(2^3 = 8)$&nbsp; </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*mit der Nebenbedingung&nbsp; $\alpha^3 =\alpha + 1$. </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Beachten Sie hinsichtlich der Codebezeichnung:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># Der dritte Parameter der für Blockcodes üblichen Nomenklatur nennt die freie Distanz&nbsp; $d_{\rm min}= 5$.<br></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># </ins>Anders als bei den im Kapitel&nbsp; [[Kanalcodierung/Beispiele_binärer_Blockcodes|<ins class="diffchange diffchange-inline">"</ins>Beispiele binärer Blockcodes<ins class="diffchange diffchange-inline">"</ins>]]&nbsp; behandelten binären Codes<ins class="diffchange diffchange-inline">&nbsp; $</ins>(<ins class="diffchange diffchange-inline">$</ins>Single Parity–check Code,<ins class="diffchange diffchange-inline">&nbsp; </ins>Repetition Code,<ins class="diffchange diffchange-inline">&nbsp; </ins>Hamming Code<ins class="diffchange diffchange-inline">$</ins>)<ins class="diffchange diffchange-inline">$&nbsp; </ins>wird bei den Reed&ndash;Solomon&ndash;Codes noch der Hinweis&nbsp; $q$&nbsp; zum Galoisfeld hinzugefügt&nbsp; $($hier: &nbsp; $q = 8)$.<br><br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Alle Elemente der Generatormatrix und der Prüfmatrix,</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Alle Elemente der Generatormatrix und der Prüfmatrix,</div></td></tr>
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<td colspan="2" class="diff-lineno">Zeile 259:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Dies soll hier nur für zwei Elemente nachgewiesen werden:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Dies soll hier nur für zwei Elemente nachgewiesen werden:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Erste Zeile, erste Spalte:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Erste Zeile,<ins class="diffchange diffchange-inline">&nbsp; </ins>erste Spalte:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>1 \hspace{0.1cm} \cdot \hspace{0.1cm} \big [1 + \alpha^1 + \alpha^2 + \alpha^3 + \alpha^4 + \alpha^5 + \alpha^6 \big ] = 1 + \alpha + \alpha^2 + (\alpha + 1) + (\alpha^2 + \alpha)+ (\alpha^2 + \alpha +1)+ (\alpha^2 + 1) = 0 \hspace{0.05cm}; </math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>1 \hspace{0.1cm} \cdot \hspace{0.1cm} \big [1 + \alpha^1 + \alpha^2 + \alpha^3 + \alpha^4 + \alpha^5 + \alpha^6 \big ] = 1 + \alpha + \alpha^2 + (\alpha + 1) + (\alpha^2 + \alpha)+ (\alpha^2 + \alpha +1)+ (\alpha^2 + 1) = 0 \hspace{0.05cm}; </math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Letzte Zeile, letzte Spalte:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Letzte Zeile,<ins class="diffchange diffchange-inline">&nbsp; </ins>letzte Spalte:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>1 \hspace{0.1cm} \cdot \hspace{0.1cm} 1 + \alpha^2 \cdot \alpha^4 + \alpha^4 \cdot \alpha^1</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>1 \hspace{0.1cm} \cdot \hspace{0.1cm} 1 + \alpha^2 \cdot \alpha^4 + \alpha^4 \cdot \alpha^1</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l263" >Zeile 263:</td>
<td colspan="2" class="diff-lineno">Zeile 272:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Eine wichtige Kenngröße eines jeden Blockcodes ist die&nbsp; [[Kanalcodierung/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung| minimale Distanz]]&nbsp; zwischen zwei beliebigen Codeworten&nbsp; $\underline {c}_i$&nbsp; und&nbsp; $\underline {c}_j$. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Eine wichtige Kenngröße eines jeden Blockcodes ist die&nbsp; [[Kanalcodierung/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung| minimale Distanz]]&nbsp; zwischen zwei beliebigen Codeworten&nbsp; $\underline {c}_i$&nbsp; und&nbsp; $\underline {c}_j$. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Die Reed&ndash;Solomon&ndash;Codes gehören zur Klasse der&nbsp; <del class="diffchange diffchange-inline"><i></del>linearen</i><del class="diffchange diffchange-inline">&nbsp; </del>und&nbsp; <del class="diffchange diffchange-inline"><i></del>zyklischen<del class="diffchange diffchange-inline"></i></del>&nbsp; Codes. Bei diesen kann man vom Nullwort&nbsp; $\underline {c}_0 = (0, 0, \hspace{0.05cm} \text{...} \hspace{0.05cm}, 0)$&nbsp; als Bezugsgröße ausgehen. </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Die Reed&ndash;Solomon&ndash;Codes gehören zur Klasse der&nbsp; <ins class="diffchange diffchange-inline">'''</ins>linearen</i> und&nbsp; zyklischen<ins class="diffchange diffchange-inline">'''</ins>&nbsp; Codes.<ins class="diffchange diffchange-inline">&nbsp; </ins>Bei diesen kann man vom Nullwort&nbsp; $\underline {c}_0 = (0, 0, \hspace{0.05cm} \text{...} \hspace{0.05cm}, 0)$&nbsp; als Bezugsgröße ausgehen. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Aus der Anzahl der Nullen in den anderen Codeworten&nbsp; $\underline {c}_j &ne; \underline {c}_0$&nbsp; lässt sich das [[Kanalcodierung/Schranken_f%C3%BCr_die_Blockfehlerwahrscheinlichkeit#Distanzspektrum_eines_linearen_Codes_.282.29| Distanzspektrum]]&nbsp; $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$&nbsp; angeben.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Aus der Anzahl der Nullen in den anderen Codeworten&nbsp; $\underline {c}_j &ne; \underline {c}_0$&nbsp; lässt sich das<ins class="diffchange diffchange-inline">&nbsp; </ins>[[Kanalcodierung/Schranken_f%C3%BCr_die_Blockfehlerwahrscheinlichkeit#Distanzspektrum_eines_linearen_Codes_.282.29| <ins class="diffchange diffchange-inline">"</ins>Distanzspektrum<ins class="diffchange diffchange-inline">"</ins>]]&nbsp; $\{ \hspace{0.05cm}W_j\hspace{0.05cm}\}$&nbsp; angeben.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l272" >Zeile 272:</td>
<td colspan="2" class="diff-lineno">Zeile 282:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Distanz&nbsp; $d$&nbsp; zwischen&nbsp; $\underline {c}_j$&nbsp; und dem Nullwort&nbsp; $\underline {c}_0$&nbsp; ist identisch dem&nbsp; [[Kanalcodierung/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung|Hamming&ndash;Gewicht]]&nbsp; $w_{\rm H}(\underline {c}_j)$.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Distanz&nbsp; $d$&nbsp; zwischen&nbsp; $\underline {c}_j$&nbsp; und dem Nullwort&nbsp; $\underline {c}_0$&nbsp; ist identisch dem&nbsp; [[Kanalcodierung/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung|Hamming&ndash;Gewicht]]&nbsp; $w_{\rm H}(\underline {c}_j)$.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P ID2520 KC T 2 3 S3 v1.png|<del class="diffchange diffchange-inline">center</del>|frame|Zur Herleitung des Distanzspektrums für den&nbsp; $\text{RSC (3, 2, 2)}_4$|class=fit]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P ID2520 KC T 2 3 S3 v1.png|<ins class="diffchange diffchange-inline">right</ins>|frame|Zur Herleitung des Distanzspektrums für den&nbsp; $\text{RSC (3, 2, 2)}_4$|class=fit]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Aus der oberen Tabelle kann unter anderem abgelesen werden:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*Neun Codeworte unterscheiden sich vom Nullwort in zwei Symbolen und sechs Codeworte in drei Symbolen: &nbsp; $W_2 = 9$,&nbsp; $W_3 = 6$. </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*Es gibt kein einziges Codewort mit nur einer Null. Das heißt: &nbsp; Die minimale Distanz beträgt hier&nbsp; $d_{\rm min} = 2$. <br></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">&rArr; &nbsp; Aus der oberen Tabelle kann unter anderem abgelesen werden:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*Neun Codeworte unterscheiden sich vom Nullwort in zwei Symbolen und sechs Codeworte in drei Symbolen: &nbsp; </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:$$W_2 = 9,\ \ W_3 = 6.$$ </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*Es gibt kein einziges Codewort mit nur einer Null. Das heißt: &nbsp; Die minimale Distanz beträgt hier&nbsp; </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:$$d_{\rm min} = 2.$$ <br></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Aus der unteren Tabelle erkennt man, dass auch für die Binärdarstellung&nbsp; $d_{\rm min} = 2$&nbsp; gilt.}}<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">&rArr; &nbsp; </ins>Aus der unteren Tabelle erkennt man, dass auch für die Binärdarstellung&nbsp; $d_{\rm min} = 2$&nbsp; gilt.}}<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Dieses empirische Ergebnis soll nun verallgemeinert werden:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Dieses empirische Ergebnis soll nun verallgemeinert werden:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l285" >Zeile 285:</td>
<td colspan="2" class="diff-lineno">Zeile 298:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Ohne Beweis:}$&nbsp; </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Ohne Beweis:}$&nbsp; </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*</del>Die&nbsp; <del class="diffchange diffchange-inline"><i></del>minimale Distanz<del class="diffchange diffchange-inline"></i></del>&nbsp; eines jeden&nbsp; $(n, k)$&ndash;Reed&ndash;Solomon&ndash;Codes ist&nbsp; $d_{\rm min} =n-k+1$ &nbsp; &rArr; &nbsp; es lassen sich&nbsp; $e = d_{\rm min} -1 =n-k$&nbsp; Symbolfehler erkennen.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># </ins>Die&nbsp; <ins class="diffchange diffchange-inline">"</ins>minimale Distanz<ins class="diffchange diffchange-inline">"</ins>&nbsp; eines jeden&nbsp; $(n, k)$&ndash;Reed&ndash;Solomon&ndash;Codes ist &nbsp; $d_{\rm min} =n-k+1$ &nbsp; &rArr; &nbsp; es lassen sich &nbsp; $e = d_{\rm min} -1 =n-k$ &nbsp; Symbolfehler erkennen.<br></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># </ins>Bei&nbsp; <ins class="diffchange diffchange-inline">"</ins>fehlerkorrigierenden Codes<ins class="diffchange diffchange-inline">"</ins>&nbsp; wählt man meist&nbsp; $d_{\rm min} $&nbsp; ungeradzahlig &nbsp; &#8658; &nbsp; $n-k$&nbsp; geradzahlig.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*</del>Bei&nbsp; <del class="diffchange diffchange-inline"><i></del>fehlerkorrigierenden Codes<del class="diffchange diffchange-inline"></i></del>&nbsp; wählt man meist&nbsp; $d_{\rm min} $&nbsp; ungeradzahlig &nbsp; &#8658; &nbsp; $n-k$&nbsp; geradzahlig. Bei <del class="diffchange diffchange-inline">RS</del>&ndash;Codes können dann bis zu&nbsp; $t =(n-k)/2$&nbsp; Symbolfehler korrigiert werden.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># </ins>Bei <ins class="diffchange diffchange-inline">Reed&ndash;Solomon</ins>&ndash;Codes können dann bis zu &nbsp; $t =(n-k)/2$&nbsp; Symbolfehler korrigiert werden.<br></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># </ins>Die&nbsp; [https://de.wikipedia.org/wiki/Singleton-Schranke Singleton&ndash;Schranke]&nbsp; besagt,<ins class="diffchange diffchange-inline">&nbsp; </ins>dass für alle linearen Codes&nbsp; $d_{\rm min} \le n-k+1$&nbsp; gilt. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*</del>Die&nbsp; [https://de.wikipedia.org/wiki/Singleton-Schranke Singleton&ndash;Schranke]&nbsp; besagt, dass für alle linearen Codes&nbsp; $d_{\rm min} \le n-k+1$&nbsp; gilt. <del class="diffchange diffchange-inline">RS</del>&ndash;Codes erreichen diese Schranke mit Gleichheit; sie sind so genannte&nbsp; [https://de.wikipedia.org/wiki/MDS-Code MDS&ndash;Codes]&nbsp; (<del class="diffchange diffchange-inline"><i></del>Maximum Distance Separable<del class="diffchange diffchange-inline"></i>&nbsp;</del>).</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># Reed&ndash;Solomon</ins>&ndash;Codes erreichen diese Schranke mit Gleichheit<ins class="diffchange diffchange-inline">;&nbsp</ins>; sie sind so genannte&nbsp; [https://de.wikipedia.org/wiki/MDS-Code MDS&ndash;Codes]&nbsp; (<ins class="diffchange diffchange-inline">"</ins>Maximum Distance Separable<ins class="diffchange diffchange-inline">"</ins>).</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># </ins>Das&nbsp; [[Kanalcodierung/Schranken_f%C3%BCr_die_Blockfehlerwahrscheinlichkeit#Distanzspektrum_eines_linearen_Codes| Distanzspektrum]]&nbsp; setzt sich zusammen aus&nbsp; $W_0 = 1$&nbsp; sowie weiteren Gewichtsfaktoren&nbsp; $W_i$&nbsp; mit&nbsp; $d &#8804; i &#8804; n$,<ins class="diffchange diffchange-inline">&nbsp; </ins>wobei in der folgenden Gleichung&nbsp; $d_{\rm min}$&nbsp; mit&nbsp; $d$&nbsp; abgekürzt ist:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*</del>Das&nbsp; [[Kanalcodierung/Schranken_f%C3%BCr_die_Blockfehlerwahrscheinlichkeit#Distanzspektrum_eines_linearen_Codes| Distanzspektrum]]&nbsp; setzt sich zusammen aus&nbsp; $W_0 = 1$&nbsp; sowie weiteren Gewichtsfaktoren&nbsp; $W_i$&nbsp; mit&nbsp; $d &#8804; i &#8804; n$, wobei in der folgenden Gleichung&nbsp; $d_{\rm min}$&nbsp; mit&nbsp; $d$&nbsp; abgekürzt ist:</div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>::<math>W_i = {n \choose i} \cdot \sum_{j = 0}^{i-d}\hspace{0.15cm}(-1)^j \cdot {i \choose j} \cdot \bigg [\hspace{0.03cm}q^{i\hspace{0.03cm}-\hspace{0.03cm}j\hspace{0.03cm}-\hspace{0.03cm}d\hspace{0.03cm}+\hspace{0.03cm}1}-1 \hspace{0.03cm} \bigg ]\hspace{0.05cm}.</math>}}</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:</ins>::<math>W_i = {n \choose i} \cdot \sum_{j = 0}^{i-d}\hspace{0.15cm}(-1)^j \cdot {i \choose j} \cdot \bigg [\hspace{0.03cm}q^{i\hspace{0.03cm}-\hspace{0.03cm}j\hspace{0.03cm}-\hspace{0.03cm}d\hspace{0.03cm}+\hspace{0.03cm}1}-1 \hspace{0.03cm} \bigg ]\hspace{0.05cm}.</math>}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Codebezeichnung und Coderate ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Codebezeichnung und Coderate ==</div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33620&oldid=prev
Guenter am 5. Oktober 2022 um 16:48 Uhr
2022-10-05T16:48:26Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 5. Oktober 2022, 16:48 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l150" >Zeile 150:</td>
<td colspan="2" class="diff-lineno">Zeile 150:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{GraueBox|TEXT= </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{GraueBox|TEXT= </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>$\text{Beispiel 3:}$&nbsp; Wir betrachten wie im&nbsp; $\text{Beispiel 2}$&nbsp; (vorherige Seite) den&nbsp; $\text{RSC (3, 2, 2)}_4$, dessen Generatormatrix folgende Form hat:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\text{Beispiel 3:}$&nbsp; Wir betrachten wie im&nbsp; <ins class="diffchange diffchange-inline">[[Kanalcodierung/Definition_und_Eigenschaften_von_Reed–Solomon–Codes#Codierprinzip_und_Codeparameter|</ins>$\text{Beispiel 2}$<ins class="diffchange diffchange-inline">]]</ins>&nbsp; (vorherige Seite)<ins class="diffchange diffchange-inline">&nbsp; </ins>den&nbsp; $\text{RSC (3, 2, 2)}_4$, dessen Generatormatrix folgende Form hat:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math> { \boldsymbol{\rm G} } = </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math> { \boldsymbol{\rm G} } = </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l161" >Zeile 161:</td>
<td colspan="2" class="diff-lineno">Zeile 161:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Daneben gilt:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Daneben gilt:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Die erste Zeile von&nbsp; $\boldsymbol{\rm G}$&nbsp; gibt das Codewort für das Informationswort&nbsp; $\underline {u}_1 = (1, 0)$&nbsp; an bzw. für die Polynomfunktion $u_1(x) = 1$. Damit erhält man die Matrixelemente der ersten Zeile zu</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Die erste Zeile von&nbsp; $\boldsymbol{\rm G}$&nbsp; gibt das Codewort für das Informationswort&nbsp; $\underline {u}_1 = (1, 0)$&nbsp; an bzw. für die Polynomfunktion<ins class="diffchange diffchange-inline">&nbsp; </ins>$u_1(x) = 1$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Damit erhält man die Matrixelemente der ersten Zeile zu</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<del class="diffchange diffchange-inline">:<math></del>g_{00} = u_{1}(\alpha^{0}) = 1\hspace{0.05cm},<del class="diffchange diffchange-inline">\hspace{0.3cm}</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<ins class="diffchange diffchange-inline">$$</ins>g_{00} = u_{1}(\alpha^{0}) = 1\hspace{0.05cm},<ins class="diffchange diffchange-inline">$$</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>g_{01} = u_{1}(\alpha^{1}) = 1\hspace{0.05cm},<del class="diffchange diffchange-inline">\hspace{0.3cm}</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:$$</ins>g_{01} = u_{1}(\alpha^{1}) = 1\hspace{0.05cm},<ins class="diffchange diffchange-inline">$$</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>g_{02} = u_{1}(\alpha^{2}) = 1\hspace{0.05cm}.<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:$$</ins>g_{02} = u_{1}(\alpha^{2}) = 1\hspace{0.05cm}.<ins class="diffchange diffchange-inline">$$</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Die zweite Zeile ist gleich dem Codewort für das Informationswort $\underline {u}_2 = (0, 1)$ &nbsp; &#8658; &nbsp; $u_2(x) = x$. Die Matrixelemente der zweiten Zeile lauten somit:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Die zweite Zeile ist gleich dem Codewort für das Informationswort<ins class="diffchange diffchange-inline">&nbsp; </ins>$\underline {u}_2 = (0,<ins class="diffchange diffchange-inline">\ </ins> 1)$ &nbsp; &#8658; &nbsp; $u_2(x) = x$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Die Matrixelemente der zweiten Zeile lauten somit:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<del class="diffchange diffchange-inline">:<math></del>g_{10} = u_{2}(\alpha^{0}) = \alpha^{0} = 1\hspace{0.05cm},<del class="diffchange diffchange-inline">\hspace{0.3cm}</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<ins class="diffchange diffchange-inline">$$</ins>g_{10} = u_{2}(\alpha^{0}) = \alpha^{0} = 1\hspace{0.05cm},<ins class="diffchange diffchange-inline">$$</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>g_{11} = u_{2}(\alpha^{1}) = \alpha \hspace{0.05cm},<del class="diffchange diffchange-inline">\hspace{0.3cm}</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:$$</ins>g_{11} = u_{2}(\alpha^{1}) = \alpha \hspace{0.05cm},<ins class="diffchange diffchange-inline">$$</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>g_{12} = u_{2}(\alpha^{2}) = \alpha^{2}\hspace{0.05cm}.<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:$$</ins>g_{12} = u_{2}(\alpha^{2}) = \alpha^{2}\hspace{0.05cm}.<ins class="diffchange diffchange-inline">$$</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>::<math><del class="diffchange diffchange-inline">\Rightarrow\hspace{0.3cm} </del>{ \boldsymbol{\rm G} } = </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">* Somit lautet die komplette Generatormatrix:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>::<math>{ \boldsymbol{\rm G} } = </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{pmatrix}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{pmatrix}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 1 & 1 & 1\\</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 1 & 1 & 1\\</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l179" >Zeile 179:</td>
<td colspan="2" class="diff-lineno">Zeile 180:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}\hspace{0.05cm}.</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}\hspace{0.05cm}.</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Für das Informationswort $\underline {u}= (u_0, \ u_1)$ mit den Symbolen $u_0, \ u_1 &#8712; \{0, \ \alpha^0, \ \alpha^1 = \alpha, \ \alpha^2\}$<del class="diffchange diffchange-inline">&nbsp; </del>erhält man unter Berücksichtigung der beiden Gleichungen&nbsp; $\alpha^2 = \alpha + 1$&nbsp; sowie&nbsp; $\alpha^3 = \alpha^0 = 1$&nbsp; wiederum die Codetabelle des&nbsp; $\text{RSC (3, 2, 2)}_4$&nbsp; auf Symbolebene.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">[[Datei:P ID2519 KC T 2 3 S2a v1.png|right|frame|Codetabelle des&nbsp; $\text{RSC (3, 2, 2)}_4$&nbsp; auf Symbolebene|class=fit]]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Für das Informationswort<ins class="diffchange diffchange-inline">&nbsp; </ins>$\underline {u}= (u_0, \ u_1)$<ins class="diffchange diffchange-inline">&nbsp; </ins>mit den Symbolen </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:$</ins>$u_0, \ u_1 &#8712; \{0, \ \alpha^0, \ \alpha^1 = \alpha, \ \alpha^2\}$<ins class="diffchange diffchange-inline">$</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>erhält man unter Berücksichtigung der beiden Gleichungen &nbsp; $\alpha^2 = \alpha + 1$ &nbsp; sowie &nbsp; $\alpha^3 = \alpha^0 = 1$ &nbsp; wiederum die Codetabelle des&nbsp; $\text{RSC (3, 2, 2)}_4$&nbsp; auf Symbolebene.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">[[Datei:P ID2519 KC T 2 3 S2a v1.png|center|frame|Codetabelle des&nbsp; $\text{RSC (3, 2, 2)}_4$&nbsp; auf Symbolebene|class=fit]]</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># </ins>Man erhält natürlich mit der <ins class="diffchange diffchange-inline">gleichen </ins>Generatormatrix genau die gleiche Codetabelle&nbsp; $\underline {u} &nbsp; &#8596; &nbsp; \underline {c}$&nbsp; wie nach der Berechnung über die Funktion&nbsp; $u(x)$. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># </ins>Die entsprechende Codetabelle auf Bitebene<ins class="diffchange diffchange-inline">&nbsp; $</ins>(<ins class="diffchange diffchange-inline">$</ins>siehe&nbsp; $\text{Beispiel 2}$&nbsp; auf der vorherigen Seite<ins class="diffchange diffchange-inline">$</ins>)<ins class="diffchange diffchange-inline">$&nbsp; </ins>ergibt sich wieder,<ins class="diffchange diffchange-inline">&nbsp; </ins>wenn man die Elemente nicht in Exponentendarstellung angibt,<ins class="diffchange diffchange-inline">&nbsp; </ins>sondern in Koeffizientenform:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*</del>Man erhält natürlich mit der Generatormatrix genau die gleiche Codetabelle&nbsp; $\underline {u} &nbsp; &#8596; &nbsp; \underline {c}$&nbsp; wie nach der Berechnung über die Funktion&nbsp; $u(x)$. </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*</del>Die entsprechende Codetabelle auf Bitebene (siehe&nbsp; $\text{Beispiel 2}$&nbsp; auf der vorherigen Seite) ergibt sich wieder, wenn man die Elemente nicht in Exponentendarstellung angibt, sondern in Koeffizientenform:</div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>(0, \hspace{0.1cm}\alpha^{0}, \hspace{0.1cm}\alpha^{1}, \hspace{0.1cm}\alpha^{2}) \hspace{0.3cm}\Leftrightarrow\hspace{0.3cm}(00, \hspace{0.1cm}01, \hspace{0.1cm}10, \hspace{0.1cm}11) </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>(0, \hspace{0.1cm}\alpha^{0}, \hspace{0.1cm}\alpha^{1}, \hspace{0.1cm}\alpha^{2}) \hspace{0.3cm}\Leftrightarrow\hspace{0.3cm}(00, \hspace{0.1cm}01, \hspace{0.1cm}10, \hspace{0.1cm}11) </div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33619&oldid=prev
Guenter am 5. Oktober 2022 um 16:35 Uhr
2022-10-05T16:35:27Z
<p></p>
<a href="//www.lntwww.de/index.php?title=Kanalcodierung/Definition_und_Eigenschaften_von_Reed%E2%80%93Solomon%E2%80%93Codes&diff=33619&oldid=27798">Änderungen zeigen</a>
Guenter