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<title>Reed-Solomon-Code</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Reed-Solomon-Code</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Reed-Solomon-Codes (kurz RS-Codes) sind eine Klasse <a href="Zyklischer_Code" title="Zyklischer Code">zyklischer</a> <a href="Blockcode" title="Blockcode">Blockcodes</a>. Sie werden im Rahmen der <a href="Kanalkodierung" title="Kanalkodierung">Kanalkodierung</a> zum Erkennen und Korrigieren von Übertragungs- oder Speicherfehlern als Teil einer <a href="Vorw%C3%A4rtsfehlerkorrektur" title="Vorwärtsfehlerkorrektur">Vorwärtsfehlerkorrektur</a> eingesetzt. Sie bilden eine Unterklasse der allgemeinen Klasse der <a href="BCH-Code" title="BCH-Code">BCH-Codes</a>. RS-Codes sind <a href="MDS-Code" title="MDS-Code">MDS-Codes</a>, womit sie im Rahmen der <a href="Kodierungstheorie" title="Kodierungstheorie">Kodierungstheorie</a> als <a href="Optimaler_Code" title="Optimaler Code">optimale Codes</a> gelten.

Reed-Solomon-Codes wurden um 1960 von <a href="Irving_S._Reed" class="mw-redirect" title="Irving S. Reed">Irving S. Reed</a> und <a href="Gustave_Solomon" title="Gustave Solomon">Gustave Solomon</a> am <a href="Lincoln_Laboratory" title="Lincoln Laboratory">Lincoln Laboratory</a>, einer Forschungseinrichtung des <a href="Verteidigungsministerium_der_Vereinigten_Staaten" title="Verteidigungsministerium der Vereinigten Staaten">Verteidigungsministeriums der Vereinigten Staaten</a> entwickelt.<a href="#cite_note-irvi1-1">[1]</a> Zu dieser Zeit war die praktische Verwendbarkeit dieser Codes allerdings eingeschränkt, da keine <a href="Effizienz_(Informatik)" title="Effizienz (Informatik)">effiziente</a> Methode zur Decodierung bekannt war. Einen effizienten Decodieralgorithmus stellten 1969 <a href="Elwyn_Berlekamp" title="Elwyn Berlekamp">Elwyn Berlekamp</a> und <a href="James_Massey" title="James Massey">James Massey</a> in Form des auch für BCH-Codes verwendbaren <a href="Berlekamp-Massey-Algorithmus" title="Berlekamp-Massey-Algorithmus">Berlekamp-Massey-Algorithmus</a> vor.

Erstmals angewandt wurden Reed-Solomon-Codes im <a href="Voyager-Programm" title="Voyager-Programm">Voyager-Programm</a> der <a href="NASA" title="NASA">NASA</a> im Jahr 1977. Erste kommerzielle Anwendung fanden sie 1982 bei der Fehlerkorrektur von <a href="Compact_Disk" class="mw-redirect" title="Compact Disk">Compact Disks</a>. Heutige Anwendungen erstrecken sich über einen großen Bereich wie den <a href="Digital_Video_Broadcasting" title="Digital Video Broadcasting">DVB</a>-Standard zur Aussendung digitaler Fernsehsignale, verschiedene <a href="Mobilfunk" title="Mobilfunk">Mobilfunkstandards</a>, <a href="Digital_Audio_Broadcasting" title="Digital Audio Broadcasting">Digital Audio Broadcasting (DAB)</a>, <a href="RAID#RAID_6:_Block-Level_Striping_mit_doppelt_verteilter_Paritätsinformation" title="RAID">RAID-6</a>-Systeme und Dateiformate wie <a href="PAR2" title="PAR2">PAR2</a> zur Datenspeicherung. Weitere Anwendungsbeispiele sind <a href="2D-Code" title="2D-Code">zweidimensionale Barcodes</a>; so setzen z. B. der <a href="QR-Code" title="QR-Code">QR-Code</a>, <a href="DataMatrix" class="mw-redirect" title="DataMatrix">DataMatrix</a>, <a href="Aztec-Code" title="Aztec-Code">Aztec-Code</a> und der <a href="PDF417" title="PDF417">PDF417</a> Reed-Solomon zur Fehlerkorrektur von Lesefehlern ein. In neueren Anwendungsbereichen werden RS-Codes zunehmend durch leistungsfähigere Codes wie die <a href="Low-Density-Parity-Check-Code" title="Low-Density-Parity-Check-Code">Low-Density-Parity-Check-Codes</a> (LDPC) oder <a href="Turbo-Code" title="Turbo-Code">Turbo-Codes</a> (TPC) abgelöst. Dies ist beispielsweise im Fernsehstandard <a href="DVB-S2" class="mw-redirect" title="DVB-S2">DVB-S2</a> der Fall, der LDPC zur Vorwärtsfehlerkorrektur einsetzt.

<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

Jede Nachricht, zum Beispiel ein Textfragment, kann als Folge aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> Zahlen kodiert und übertragen werden. Ein typisches Beispiel für die Kodierung von Texten ist der <a href="American_Standard_Code_for_Information_Interchange" title="American Standard Code for Information Interchange">ASCII-Standard</a>. Wird eine kodierte Nachricht von einem Sender zu einem Empfänger übertragen, besteht die Gefahr, dass es zu <a href="Bitfehler" title="Bitfehler">Übertragungsfehlern</a> kommt. Das bedeutet, dass einige Zahlen der Nachricht ausgelöscht oder verfälscht werden. Der Empfänger der Nachricht hat keine Möglichkeit, den Übertragungsfehler zu bemerken, da man einer Zahl per se nicht ansehen kann, ob sie richtig oder falsch ist. Einem Übertragungsfehler kann aber auf Sender-Seite entgegengewirkt werden, indem zusätzlich zur Nachricht redundante Informationen übertragen werden. Der Empfänger kann dann durch den Vergleich der erhaltenen Nachricht mit den redundant übertragenen Informationen nicht nur die Integrität der übertragenen Nachricht verifizieren, sondern zusätzlich erkannte Fehler in der Nachricht korrigieren.

Um Redundanz zur Nachricht hinzuzufügen, werden die Zahlen der Nachricht als Werte eines Polynoms an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle k-1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21363ebd7038c93aae93127e7d910fc1b2e2c745.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}" loading="lazy"> oder kleiner kann als Summe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> <a href="Monom" title="Monom">Monomen</a> dargestellt werden. Die <a href="Koeffizient" title="Koeffizient">Koeffizienten</a> dieser Monome ergeben sich als Lösung eines <a href="Lineares_Gleichungssystem" title="Lineares Gleichungssystem">linearen Gleichungssystems</a>. Aufgrund der speziellen Form dieses Systems gibt es eine Lösungsformel, die Lagrange-<a href="Polynominterpolation" title="Polynominterpolation">Interpolation</a>. Das so erhaltene Polynom wird auf weitere Stützstellen <a href="Extrapoliert" class="mw-redirect" title="Extrapoliert">extrapoliert</a>, sodass die kodierte Nachricht insgesamt aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>k}">

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<annotation encoding="application/x-tex">{\displaystyle n>k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8afbc0693bee3f48a31d2c991ddc8b6b4a35322.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.704ex; height:2.176ex;" alt="{\displaystyle n>k}" loading="lazy"> Zahlen besteht.

Werden bei der Übertragung nun einige wenige Zahlen ausgelöscht, sodass immer noch mehr als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle n-k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" loading="lazy">) korrigiert werden wie Verfälschungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-k)/2}">

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<annotation encoding="application/x-tex">{\displaystyle (n-k)/2}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4e64a932c23c9c478ad0948a57c3496ade18057.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.581ex; height:2.843ex;" alt="{\displaystyle (n-k)/2}" loading="lazy">, daher führen Lesesysteme, die Auslöschungen beim Empfang der Nachricht erkennen und mit den <a href="Nutzdaten" title="Nutzdaten">Nutzdaten</a> ausgeben, in der Regel zu einer verbesserten Korrekturfähigkeit.

Die in der <a href="Interpolation_(Mathematik)" title="Interpolation (Mathematik)">Interpolation</a> auftretenden Ausdrücke enthalten Divisionen, müssen also über einem <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> durchgeführt werden. Werden die Zahlen – oder Symbole – der Nachricht aus den <a href="Ganze_Zahl" title="Ganze Zahl">ganzen Zahlen</a> gewählt, so finden die Rechnungen also in den <a href="Rationale_Zahl" title="Rationale Zahl">rationalen Zahlen</a> statt. Außerdem können die <a href="Extrapolation" title="Extrapolation">extrapolierten</a> Werte sehr groß werden, was eventuell im vorliegenden Übertragungskanal nicht übermittelt werden kann. Um diese Nachteile zu beheben, führt man die Rechnungen in einem <a href="Endlicher_K%C3%B6rper" title="Endlicher Körper">endlichen Körper</a> durch. Dieser hat eine endliche Anzahl von Elementen, die durchnummeriert werden können, um sie mit den Symbolen der Nachricht zu verknüpfen. Die Division – außer durch Null – ist uneingeschränkt durchführbar, und somit auch die Interpolation.

Reed-Solomon-Codes sind zur Korrektur von <a href="Burstfehler" title="Burstfehler">Burstfehlern</a> bei der Datenübertragung geeignet. Bei Burstfehlern erscheinen fehlerhafte („gekippte“) Bits häufig als eine zusammenhängende Kette von Fehlern im <a href="Datenstrom" title="Datenstrom">Datenstrom</a>. Beispielsweise werden durch einen Kratzer auf einer CD mit jeder Umdrehung viele aufeinanderfolgende Bits nicht richtig gelesen. Bei der CD werden die Daten allerdings noch <a href="Cross-interleaved_Reed-Solomon_Code" title="Cross-interleaved Reed-Solomon Code">verschränkt</a>, damit die Korrekturfähigkeit auch bei Burstfehlern möglichst hoch bleibt.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d35035371db7bee93733c68c1802114c17d8bb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.479ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{p}}" loading="lazy"> ein <a href="Endlicher_K%C3%B6rper" title="Endlicher Körper">endlicher Körper</a> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> Elementen (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=q^{m}}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle p=q^{m}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b27d80c49bc1bd3970ab7b6cfa22b4b90c87582.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.112ex; height:2.676ex;" alt="{\displaystyle p=q^{m}}" loading="lazy"> ist dann notwendigerweise eine <a href="Primzahl" title="Primzahl">Primzahlpotenz</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> prim). Es werden nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> <a href="Paarweise_verschieden" title="Paarweise verschieden">paarweise verschiedene</a> Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1},\dots ,u_{n}\in \mathbb {F} _{p}}">

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<annotation encoding="application/x-tex">{\displaystyle u_{1},\dots ,u_{n}\in \mathbb {F} _{p}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3c733824a096ff8391f54719cd0ae52999b926b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.43ex; height:2.843ex;" alt="{\displaystyle u_{1},\dots ,u_{n}\in \mathbb {F} _{p}}" loading="lazy"> ausgewählt und fixiert.

Die Menge der Kodewörter eines Reed-Solomon-Codes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{RS}}(p,k,n)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{RS}}(p,k,n)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82ce3ecca13bcf67974c8dd24db639d1898d59ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.656ex; height:2.843ex;" alt="{\displaystyle {\text{RS}}(p,k,n)}" loading="lazy"> der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> für Nachrichten der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d35035371db7bee93733c68c1802114c17d8bb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.479ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{p}}" loading="lazy"> ergibt sich nun durch die Wertetupel aller <a href="Polynom" title="Polynom">Polynome</a> aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}[x]}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}[x]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f962a9c3f2d3e2d1a8dafefddfbc7d4f4ecb386a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.103ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{p}[x]}" loading="lazy"> mit Grad kleiner <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle C=\left\{a=(a_{1},\dots ,a_{n})\in \mathbb {F} _{p}{}^{n}\;{\Big |}\;a_{j}=f(u_{j}),\;j=1,\dots ,n\right\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/287a120977f04aef18e7679d3f05af9cce3b3e9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.526ex; height:4.843ex;" alt="{\displaystyle C=\left\{a=(a_{1},\dots ,a_{n})\in \mathbb {F} _{p}{}^{n}\;{\Big |}\;a_{j}=f(u_{j}),\;j=1,\dots ,n\right\}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \mathbb {F} _{p}[x]}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f\in \mathbb {F} _{p}[x]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af88978d24ef30773f712590df7f0d2d48a1ec20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.222ex; height:3.009ex;" alt="{\displaystyle f\in \mathbb {F} _{p}[x]}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(f)<k}">

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<annotation encoding="application/x-tex">{\displaystyle \deg(f)<k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0618815338362cf9d1e4fc43935d7303c148d196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.885ex; height:2.843ex;" alt="{\displaystyle \deg(f)<k}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Stützstellenmengen">Stützstellenmengen</h3></div>

RS-Codes zu verschiedenen zulässigen Stützstellenmengen sind linear isomorph. Die <a href="Bijektive_Funktion" title="Bijektive Funktion">bijektive</a> <a href="Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildung</a>, die die Isomorphie vermittelt, ergibt sich durch <a href="Lagrange-Interpolation" class="mw-redirect" title="Lagrange-Interpolation">Lagrange-Interpolation</a> bezüglich der ersten Stützstellenmenge und Auswertung in der zweiten Stützstellenmenge. Dabei werden im ersten Schritt Kodewörter in Polynome kleiner <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {F} _{p}}">

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {F} _{p}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c243e5dbcbc635fd7682d2e7f4a42f7ed697cd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.808ex; height:2.843ex;" alt="{\displaystyle \alpha \in \mathbb {F} _{p}}" loading="lazy"> ein Element der Ordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{1}=1,\,u_{2}=\alpha ,\,\dots ,u_{j}=\alpha ^{j-1},\dots ,u_{n}=\alpha ^{n-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e621913e27ce32dfaa5695aa289f23a8a3acefeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.068ex; height:3.343ex;" alt="{\displaystyle u_{1}=1,\,u_{2}=\alpha ,\,\dots ,u_{j}=\alpha ^{j-1},\dots ,u_{n}=\alpha ^{n-1}}" loading="lazy"></dd></dl>

gewählt werden. Jeder endliche Körper enthält ein <a href="Erzeuger_einer_Gruppe" class="mw-redirect" title="Erzeuger einer Gruppe">erzeugendes</a> oder <a href="Primitivwurzel" title="Primitivwurzel">primitives Element</a> der multiplikativen Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}{}^{*}=\mathbb {F} _{p}\setminus \{0\}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}{}^{*}=\mathbb {F} _{p}\setminus \{0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67a93559fbdd53632ddf6867e1facc2af81a26dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.794ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{p}{}^{*}=\mathbb {F} _{p}\setminus \{0\}}" loading="lazy">, das heißt ein Element der Ordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f356ae51988add41a7da343e6b6d48fa968da162.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p-1}" loading="lazy">. Daher ist diese spezielle Wahl für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=p-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=p-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/693ddc903047f82302c3c24a5b7ab77daa06e615.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.665ex; height:2.509ex;" alt="{\displaystyle n=p-1}" loading="lazy"> immer möglich.

Sind die Stützstellen genau die Potenzen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}=1,\;u_{j}=\alpha ^{j-1}\neq 1,\;j=2,\dots ,n,}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{1}=1,\;u_{j}=\alpha ^{j-1}\neq 1,\;j=2,\dots ,n,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dab2212b24df19a8445786e0a9694c1877838a71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.538ex; height:3.343ex;" alt="{\displaystyle u_{1}=1,\;u_{j}=\alpha ^{j-1}\neq 1,\;j=2,\dots ,n,}" loading="lazy"> eines Elementes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {F} _{p}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {F} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c243e5dbcbc635fd7682d2e7f4a42f7ed697cd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.808ex; height:2.843ex;" alt="{\displaystyle \alpha \in \mathbb {F} _{p}}" loading="lazy"> der Ordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ^{n}=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{n}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dcac1d24d0ceb7bff6fa8da24f9e86327002ca96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.967ex; height:2.343ex;" alt="{\displaystyle \alpha ^{n}=1}" loading="lazy">, so ist der RS-Kode ein <a href="Zyklischer_Code" title="Zyklischer Code">zyklischer Code</a>. Denn das Kodewort zum <a href="Polynom" title="Polynom">Polynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{j}(x)=f(\alpha ^{j}x)}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{j}(x)=f(\alpha ^{j}x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0abd5091103940cda580c7954266b4506c6dfa43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.101ex; height:3.343ex;" alt="{\displaystyle f_{j}(x)=f(\alpha ^{j}x)}" loading="lazy"> ergibt sich durch Rotation des Kodewortes zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> um <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> Stellen nach links. Wegen der einfacheren Implementierbarkeit zyklischer Codes wird diese Variante im Allgemeinen bevorzugt.

<div class="mw-heading mw-heading3"><h3 id="Kodieren_von_Nachrichten">Kodieren von Nachrichten</h3></div>

Man kann eine Nachricht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2},\dots ,a_{k})\in \mathbb {F} _{p}{}^{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2},\dots ,a_{k})\in \mathbb {F} _{p}{}^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2528d2450eacb90710008ca0ea6da9bc71a56b33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.317ex; height:3.343ex;" alt="{\displaystyle (a_{1},a_{2},\dots ,a_{k})\in \mathbb {F} _{p}{}^{k}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=a_{1}+a_{2}\,x+a_{3}\,x^{2}+\dots +a_{k}\,x^{k-1}=\sum _{i=1}^{k}a_{i}\,x^{i-1}\in \mathbb {F} _{p}[x]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/615093a8bc4368425cfddaf9cceae621951e834c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.596ex; height:7.343ex;" alt="{\displaystyle f(x)=a_{1}+a_{2}\,x+a_{3}\,x^{2}+\dots +a_{k}\,x^{k-1}=\sum _{i=1}^{k}a_{i}\,x^{i-1}\in \mathbb {F} _{p}[x]}" loading="lazy"></dd></dl>

einsetzt und dieses an den Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1},u_{2},\dots ,u_{n}\in \{0,1,...,p-1\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{1},u_{2},\dots ,u_{n}\in \{0,1,...,p-1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14d633e66edb64bfd0db748987c7dcb6fc370d0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.395ex; height:2.843ex;" alt="{\displaystyle u_{1},u_{2},\dots ,u_{n}\in \{0,1,...,p-1\}}" loading="lazy"> auswertet. Es ergibt sich damit ein Kodewort

<msub>

</mrow>

</msub>

<msub>

</mrow>

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<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=(c_{1},c_{2},\dots ,c_{n})={\Big (}f(u_{1}),f(u_{2}),\dots ,f(u_{n}){\Big )}\in \mathbb {F} _{p}{}^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e9ae290fd464adb1f9e7c7963e2338ac54f1404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.679ex; height:4.843ex;" alt="{\displaystyle c=(c_{1},c_{2},\dots ,c_{n})={\Big (}f(u_{1}),f(u_{2}),\dots ,f(u_{n}){\Big )}\in \mathbb {F} _{p}{}^{n}}" loading="lazy"></dd></dl>

der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

Für die Anzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\leq n-d+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\leq n-d+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f09cbed3e5d170bbfb30b74fc1c145aa2932eb09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.764ex; height:2.343ex;" alt="{\displaystyle k\leq n-d+1}" loading="lazy">. Weil ein beliebiges <a href="Polynom" title="Polynom">Polynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> vom Grad kleiner oder gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> Symbole enthält, die ungleich 0 sind. Daher hat der gebildete Code die Minimaldistanz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> und ist in der Lage, maximal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\frac {d-1}{2}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t={\frac {d-1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b6a24952501f6289aefd15ffaf0c6c41f91e143.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.993ex; height:5.343ex;" alt="{\displaystyle t={\frac {d-1}{2}}}" loading="lazy"> Fehler zu korrigieren.<a href="#cite_note-:0-2">[2]</a>

Statt die Nachricht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},\dots ,a_{k})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{1},\dots ,a_{k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e87a13198a4f29a6523f2ada1f8d64dd175c505b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.59ex; height:2.843ex;" alt="{\displaystyle (a_{1},\dots ,a_{k})}" loading="lazy"> als Polynomkoeffizienten zu kodieren, kann man sie alternativ auch in die ersten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> ergibt sich dabei als <a href="Polynominterpolation#Lagrangesche_Interpolationsformel" title="Polynominterpolation">Lagrange-Polynom</a>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{i=1}^{k}\left(a_{i}\cdot \prod _{j\neq i}^{k}{\frac {x-u_{j}}{u_{i}-u_{j}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cfd5b16254e5348c8620a20b0f74dea155682fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:30.05ex; height:7.843ex;" alt="{\displaystyle f(x)=\sum _{i=1}^{k}\left(a_{i}\cdot \prod _{j\neq i}^{k}{\frac {x-u_{j}}{u_{i}-u_{j}}}\right)}" loading="lazy">

der Paare <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Big (}(u_{1},a_{1}),\,(u_{2},a_{2}),\,\ldots ,\,(u_{k},a_{k}){\Big )}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\Big (}(u_{1},a_{1}),\,(u_{2},a_{2}),\,\ldots ,\,(u_{k},a_{k}){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a44c40eee0bbafe1ed1be127cbf6870063e9b87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.752ex; height:4.843ex;" alt="{\displaystyle {\Big (}(u_{1},a_{1}),\,(u_{2},a_{2}),\,\ldots ,\,(u_{k},a_{k}){\Big )}}" loading="lazy">. Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u_{i})=a_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(u_{i})=a_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1eb2f2e9df73267a581250a434d7d61071b7e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.345ex; height:2.843ex;" alt="{\displaystyle f(u_{i})=a_{i}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dots ,k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=1,\dots ,k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a0955763d859a60b1d90c7287a7c566e9c82d66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.453ex; height:2.509ex;" alt="{\displaystyle i=1,\dots ,k}" loading="lazy"> ergibt sich aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle c=(c_{1},c_{2},\dots ,c_{n})={\Big (}a_{1},a_{2},\ldots ,a_{k},f(u_{k+1}),\dots ,f(u_{n}){\Big )}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02f36371614102c45d87d8c238cc13f64fc5bc57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.902ex; height:4.843ex;" alt="{\displaystyle c=(c_{1},c_{2},\dots ,c_{n})={\Big (}a_{1},a_{2},\ldots ,a_{k},f(u_{k+1}),\dots ,f(u_{n}){\Big )}}" loading="lazy"></dd></dl>

mit der Nachricht in den ersten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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Beide Varianten benutzen dieselbe Menge von Kodewörtern und haben damit dieselben Fehlerkorrektureigenschaften.

Aus den Koeffizienten des <a href="Polynom" title="Polynom">Polynoms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=a_{1}+a_{2}\,x+a_{3}\,x^{2}+\dots +a_{k}\,x^{k-1}}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=a_{1}+a_{2}\,x+a_{3}\,x^{2}+\dots +a_{k}\,x^{k-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24e0e50fb3e7772f4cfe547b7282dcbf560061ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.165ex; height:3.176ex;" alt="{\displaystyle f(x)=a_{1}+a_{2}\,x+a_{3}\,x^{2}+\dots +a_{k}\,x^{k-1}}" loading="lazy"> erhält man die Erzeugendenmatrix für den Reed-Solomon-Code:<a href="#cite_note-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle G={\begin{pmatrix}a_{1}&a_{2}&\ldots &a_{k}&1&0&\ldots &0\\0&a_{1}&a_{2}&\ldots &a_{k}&1&\ddots &\vdots \\\vdots &\ddots &\ddots &\ddots &\ddots &\ddots &\ddots &0\\0&\ldots &0&a_{1}&a_{2}&\ldots &a_{k}&1\\\end{pmatrix}}\in M(k\times n,\mathbb {F} _{p})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/539fb5b55b45c436b88fae7d7eb1335c54c62128.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:63.372ex; height:15.843ex;" alt="{\displaystyle G={\begin{pmatrix}a_{1}&a_{2}&\ldots &a_{k}&1&0&\ldots &0\\0&a_{1}&a_{2}&\ldots &a_{k}&1&\ddots &\vdots \\\vdots &\ddots &\ddots &\ddots &\ddots &\ddots &\ddots &0\\0&\ldots &0&a_{1}&a_{2}&\ldots &a_{k}&1\\\end{pmatrix}}\in M(k\times n,\mathbb {F} _{p})}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

Durch die Definition ergeben sich sofort folgende Eigenschaften:

<ul><li>Codewortlänge: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<li><a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimension</a> des Codes: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |C|=|f|=q^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle |C|=|f|=q^{k}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6b40798026b1f4f48db3bbdc7280b6c320c909d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.997ex; height:3.176ex;" alt="{\displaystyle |C|=|f|=q^{k}}" loading="lazy"></li>

<li>Coderate: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{c}=k/n}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd9e74de29a38e13101275847cd6a1eabfffa1b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.575ex; height:2.843ex;" alt="{\displaystyle R_{c}=k/n}" loading="lazy"></li></ul>

Die Mindestdistanz beträgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\text{min}}=n-k+1}">

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<annotation encoding="application/x-tex">{\displaystyle d_{\text{min}}=n-k+1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f8d6edc62337bedade677eb97b8e3a3ba3aaf8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.729ex; height:2.509ex;" alt="{\displaystyle d_{\text{min}}=n-k+1}" loading="lazy"> und erfüllt damit die <a href="Singleton-Schranke" title="Singleton-Schranke">Singleton-Schranke</a>. Codes mit dieser Eigenschaft werden auch <a href="MDS-Code" title="MDS-Code">MDS-Codes</a> genannt.

<dl><dt>Erklärung</dt>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> maximal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21363ebd7038c93aae93127e7d910fc1b2e2c745.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}" loading="lazy"> <a href="Nullstelle" title="Nullstelle">Nullstellen</a> besitzen kann (durch den Grad des <a href="Polynom" title="Polynom">Polynoms</a> beschränkt), tauchen im korrespondierenden Codewort maximal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21363ebd7038c93aae93127e7d910fc1b2e2c745.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}" loading="lazy"> Stellen auf, die zu 0 werden. Damit ist das <a href="Hamming-Gewicht" title="Hamming-Gewicht">Hamming-Gewicht</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle wt(C)\geqq n-k+1}">

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<annotation encoding="application/x-tex">{\displaystyle wt(C)\geqq n-k+1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c49b339639795db0ba6824b554db64595709974.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.627ex; height:2.843ex;" alt="{\displaystyle wt(C)\geqq n-k+1}" loading="lazy"> und somit wegen der <a href="Linearit%C3%A4t_(Mathematik)" title="Linearität (Mathematik)">Linearität</a> auch die Minimaldistanz.</dd>

<dd>Zusammen mit der <a href="Singleton-Schranke" title="Singleton-Schranke">Singleton-Schranke</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\text{min}}\leqq n-k+1}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42783749916046a10899fe16c69822868160cdd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.729ex; height:2.676ex;" alt="{\displaystyle d_{\text{min}}\leqq n-k+1}" loading="lazy"> ergibt sich die Gleichheit.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>

Gegeben ist die Nachricht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})=(2,6,8,12,15,13,1)}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5eb247ade73e5a2e2117b499fffb2880f1c91bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.419ex; height:2.843ex;" alt="{\displaystyle (a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})=(2,6,8,12,15,13,1)}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2^{4}}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cc1c5ba8fd18c73ae54aaeacdf39d60d0cc8722.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.306ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{2^{4}}}" loading="lazy">. Daraus erhält man das <a href="Polynom" title="Polynom">Polynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=2+6\cdot x+8\cdot x^{2}+12\cdot x^{3}+15\cdot x^{4}+13\cdot x^{5}+x^{6}}">

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<th>Exponentendarstellung

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<th>Komponentendarstellung

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<th>binäre Darstellung

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<th>dezimale Darstellung

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [1000]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71f4bb946322794b92e7f4bda0a3dbf41714ac3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [1000]_{2}}" loading="lazy">

</td>

<td>8

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{4}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9da787f0e603ea817720825f39fac6a8fa2df07c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.676ex;" alt="{\displaystyle \alpha ^{4}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{3}\quad \quad \quad \quad +\alpha ^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a24c88dec86e6d59e2db3a1f78a94c955660d13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.214ex; height:2.843ex;" alt="{\displaystyle \alpha ^{3}\quad \quad \quad \quad +\alpha ^{0}}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [1001]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbad144b1895737009e9bc27767cf4c3acc6a74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [1001]_{2}}" loading="lazy">

</td>

<td>9

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{5}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1e48f2de64a91c8aebb961411be6b8c0697624e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.676ex;" alt="{\displaystyle \alpha ^{5}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{3}\quad \quad +\alpha ^{1}+\alpha ^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24ed231d5355a61a1a0f4b3d38938944e74a5c84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.952ex; height:2.843ex;" alt="{\displaystyle \alpha ^{3}\quad \quad +\alpha ^{1}+\alpha ^{0}}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [1011]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/238d5d00e81f3c1e32cfa806509f6e4dd7d90b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [1011]_{2}}" loading="lazy">

</td>

<td>11

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{6}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c3240fcedfef5951390c6bc3f83a776d6100e0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.676ex;" alt="{\displaystyle \alpha ^{6}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{3}+\alpha ^{2}+\alpha ^{1}+\alpha ^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/384d6498fd7ab091312cc0a3f53e072eabad0f62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.689ex; height:2.843ex;" alt="{\displaystyle \alpha ^{3}+\alpha ^{2}+\alpha ^{1}+\alpha ^{0}}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [1111]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ea95e4a380ec064c4ac7c61b2cbad10716392fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [1111]_{2}}" loading="lazy">

</td>

<td>15

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{7}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bf1ea7e7e6bda01513bfcde2b22b3e7465e4f1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.676ex;" alt="{\displaystyle \alpha ^{7}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad \quad \alpha ^{2}+\alpha ^{1}+\alpha ^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7771872968f5ddd000404f7b0b93cccb26ad930f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.952ex; height:2.843ex;" alt="{\displaystyle \quad \quad \alpha ^{2}+\alpha ^{1}+\alpha ^{0}}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0111]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbd2ae105ae386e52325e436a3edb625de1d2908.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [0111]_{2}}" loading="lazy">

</td>

<td>7

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{8}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5417c6d4f94536fe2b64019e310c111415dbb440.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.676ex;" alt="{\displaystyle \alpha ^{8}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{3}+\alpha ^{2}+\alpha ^{1}\quad \quad }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81d0374b8f199e65ccebe796580d0eecb0165924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.952ex; height:2.843ex;" alt="{\displaystyle \alpha ^{3}+\alpha ^{2}+\alpha ^{1}\quad \quad }" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [1110]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06e22991f80b3685048b3226c35b292c7b5d705d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [1110]_{2}}" loading="lazy">

</td>

<td>14

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{9}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9578d665091e41730986f09c73b15b9ab63e795c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:2.676ex;" alt="{\displaystyle \alpha ^{9}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad \quad \alpha ^{2}\quad \quad +\alpha ^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48e521acaa33ef191a91f92c9da45eb521e0730f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.214ex; height:2.843ex;" alt="{\displaystyle \quad \quad \alpha ^{2}\quad \quad +\alpha ^{0}}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0101]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0b57ea39fdcacc34de043628d6d6a957dcc7d02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [0101]_{2}}" loading="lazy">

</td>

<td>5

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{10}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc97ca679ffafabbd3d67ec2939f0d71bb80725f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.364ex; height:2.676ex;" alt="{\displaystyle \alpha ^{10}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{3}\quad \quad +\alpha ^{1}\quad \quad }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06284e4b5c5f3867defd5aafad7874e107c13134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.214ex; height:2.843ex;" alt="{\displaystyle \alpha ^{3}\quad \quad +\alpha ^{1}\quad \quad }" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [1010]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e35c1beb2c3f166e39079b5e2e79800a3f15f53f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [1010]_{2}}" loading="lazy">

</td>

<td>10

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{11}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3833dc081dc4e40f3360315a75161f5d5e4be3a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.364ex; height:2.676ex;" alt="{\displaystyle \alpha ^{11}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{3}+\alpha ^{2}\quad \quad +\alpha ^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1fca3ccf9e7a639a3bfe3c790d6a4085623d325.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.952ex; height:2.843ex;" alt="{\displaystyle \alpha ^{3}+\alpha ^{2}\quad \quad +\alpha ^{0}}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [1101]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0ab9c4b9bdce6d0f83482089098f1a2a906811.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [1101]_{2}}" loading="lazy">

</td>

<td>13

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{12}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ddf5aabd3ff46616cda88e9c580c48bf3f565524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.364ex; height:2.676ex;" alt="{\displaystyle \alpha ^{12}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad \quad \quad \quad \alpha ^{1}+\alpha ^{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41763bc9867f88fe8a624b5b3570cf293ce4bd4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.214ex; height:2.843ex;" alt="{\displaystyle \quad \quad \quad \quad \alpha ^{1}+\alpha ^{0}}" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0011]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b93d32776fbd4caf156de7b46612089f51b064c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [0011]_{2}}" loading="lazy">

</td>

<td>3

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{13}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b3704de645a783e1825b813889e8cf01e7ac197.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.364ex; height:2.676ex;" alt="{\displaystyle \alpha ^{13}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad \quad \alpha ^{2}+\alpha ^{1}\quad \quad }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7079043bd4067aae70611b4926f0ee8f5a4f4f7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.214ex; height:2.843ex;" alt="{\displaystyle \quad \quad \alpha ^{2}+\alpha ^{1}\quad \quad }" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0110]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e83c8f312bc2a4cf2975ff80aec34922c770e7bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [0110]_{2}}" loading="lazy">

</td>

<td>6

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{14}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c6cf1f4d93f99d6d9e9d11c2742a84ee7e75fdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.364ex; height:2.676ex;" alt="{\displaystyle \alpha ^{14}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ^{3}+\alpha ^{2}\quad \quad \quad \quad }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e5d73f5d0a2451f53d721ca6013d17f318fc9d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.214ex; height:2.843ex;" alt="{\displaystyle \alpha ^{3}+\alpha ^{2}\quad \quad \quad \quad }" loading="lazy">

</td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [1100]_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b72be4c048a991bc8e297a6ac8e4c620db80c748.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.998ex; height:2.843ex;" alt="{\displaystyle [1100]_{2}}" loading="lazy">

</td>

<td>12

</td></tr></tbody></table>

Daraus ergeben sich die Werte über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2^{4}}}">

<msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2^{4}}}</annotation>

</semantics>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c_{1}=f(\alpha ^{0})&=2\cdot \alpha ^{0}+6\cdot \alpha ^{0}+8\cdot \alpha ^{0}+12\cdot \alpha ^{0}+15\cdot \alpha ^{0}+13\cdot \alpha ^{0}+\alpha ^{0}\\&=2+6+8+12+15+13+1\\&=[0010]_{2}+[0110]_{2}+[1000]_{2}+[1100]_{2}+[1111]_{2}+[1101]_{2}+[0001]_{2}\\&=[0011]_{2}=3\\c_{2}=f(\alpha ^{1})&=2\cdot \alpha ^{0}+6\cdot \alpha ^{1}+8\cdot \alpha ^{2}+12\cdot \alpha ^{3}+15\cdot \alpha ^{4}+13\cdot \alpha ^{5}+\alpha ^{6}\\&=\alpha ^{1}\cdot \alpha ^{0}+\alpha ^{13}\cdot \alpha ^{1}+\alpha ^{3}\cdot \alpha ^{2}+\alpha ^{14}\cdot \alpha ^{3}+\alpha ^{6}\cdot \alpha ^{4}+\alpha ^{11}\cdot \alpha ^{5}+\alpha ^{0}\cdot \alpha ^{6}\\&=\alpha ^{1}+\alpha ^{14}+\alpha ^{5}+\alpha ^{2}+\alpha ^{10}+\alpha ^{1}\cdot \alpha ^{6}\\&=2+12+11+4+10+2+15\\&=[0010]_{2}+[1100]_{2}+[1011]_{2}+[0100]_{2}+[1010]_{2}+[0010]_{2}+[1111]_{2}\\&=[0110]_{2}=6\\&\cdots \\c_{15}=f(\alpha ^{14})&=2\cdot \alpha ^{0}+6\cdot \alpha ^{14}+8\cdot \alpha ^{28}+12\cdot \alpha ^{42}+15\cdot \alpha ^{56}+13\cdot \alpha ^{70}+\alpha ^{84}\\&=\alpha ^{1}\cdot \alpha ^{0}+\alpha ^{13}\cdot \alpha ^{14}+\alpha ^{3}\cdot \alpha ^{13}+\alpha ^{14}\cdot \alpha ^{12}+\alpha ^{6}\cdot \alpha ^{11}+\alpha ^{11}\cdot \alpha ^{10}+\alpha ^{0}\cdot \alpha ^{9}\\&=\alpha ^{1}+\alpha ^{12}+\alpha ^{1}+\alpha ^{11}+\alpha ^{2}+\alpha ^{6}\cdot \alpha ^{9}\\&=2+3+2+13+4+15+5\\&=[0010]_{2}+[0011]_{2}+[0010]_{2}+[1101]_{2}+[0100]_{2}+[1111]_{2}+[0101]_{2}\\&=[0000]_{2}=0\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03dcc7b8d26d88d47ba34cce40b8d3d90532d798.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -26.338ex; width:87.241ex; height:53.676ex;" alt="{\displaystyle {\begin{aligned}c_{1}=f(\alpha ^{0})&=2\cdot \alpha ^{0}+6\cdot \alpha ^{0}+8\cdot \alpha ^{0}+12\cdot \alpha ^{0}+15\cdot \alpha ^{0}+13\cdot \alpha ^{0}+\alpha ^{0}\\&=2+6+8+12+15+13+1\\&=[0010]_{2}+[0110]_{2}+[1000]_{2}+[1100]_{2}+[1111]_{2}+[1101]_{2}+[0001]_{2}\\&=[0011]_{2}=3\\c_{2}=f(\alpha ^{1})&=2\cdot \alpha ^{0}+6\cdot \alpha ^{1}+8\cdot \alpha ^{2}+12\cdot \alpha ^{3}+15\cdot \alpha ^{4}+13\cdot \alpha ^{5}+\alpha ^{6}\\&=\alpha ^{1}\cdot \alpha ^{0}+\alpha ^{13}\cdot \alpha ^{1}+\alpha ^{3}\cdot \alpha ^{2}+\alpha ^{14}\cdot \alpha ^{3}+\alpha ^{6}\cdot \alpha ^{4}+\alpha ^{11}\cdot \alpha ^{5}+\alpha ^{0}\cdot \alpha ^{6}\\&=\alpha ^{1}+\alpha ^{14}+\alpha ^{5}+\alpha ^{2}+\alpha ^{10}+\alpha ^{1}\cdot \alpha ^{6}\\&=2+12+11+4+10+2+15\\&=[0010]_{2}+[1100]_{2}+[1011]_{2}+[0100]_{2}+[1010]_{2}+[0010]_{2}+[1111]_{2}\\&=[0110]_{2}=6\\&\cdots \\c_{15}=f(\alpha ^{14})&=2\cdot \alpha ^{0}+6\cdot \alpha ^{14}+8\cdot \alpha ^{28}+12\cdot \alpha ^{42}+15\cdot \alpha ^{56}+13\cdot \alpha ^{70}+\alpha ^{84}\\&=\alpha ^{1}\cdot \alpha ^{0}+\alpha ^{13}\cdot \alpha ^{14}+\alpha ^{3}\cdot \alpha ^{13}+\alpha ^{14}\cdot \alpha ^{12}+\alpha ^{6}\cdot \alpha ^{11}+\alpha ^{11}\cdot \alpha ^{10}+\alpha ^{0}\cdot \alpha ^{9}\\&=\alpha ^{1}+\alpha ^{12}+\alpha ^{1}+\alpha ^{11}+\alpha ^{2}+\alpha ^{6}\cdot \alpha ^{9}\\&=2+3+2+13+4+15+5\\&=[0010]_{2}+[0011]_{2}+[0010]_{2}+[1101]_{2}+[0100]_{2}+[1111]_{2}+[0101]_{2}\\&=[0000]_{2}=0\end{aligned}}}" loading="lazy">

und das Kodewort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=(c_{1},c_{2},\ldots ,c_{15})={\Big (}f(\alpha ^{0}),f(\alpha ^{1}),\ldots ,f(\alpha ^{14}){\Big )}=(3,6,15,6,6,3,13,14,3,12,15,2,11,1,0)}">

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<annotation encoding="application/x-tex">{\displaystyle c=(c_{1},c_{2},\ldots ,c_{15})={\Big (}f(\alpha ^{0}),f(\alpha ^{1}),\ldots ,f(\alpha ^{14}){\Big )}=(3,6,15,6,6,3,13,14,3,12,15,2,11,1,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a9205f10670a153fa73dfa9374f0d5f5f68ad2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:92.724ex; height:4.843ex;" alt="{\displaystyle c=(c_{1},c_{2},\ldots ,c_{15})={\Big (}f(\alpha ^{0}),f(\alpha ^{1}),\ldots ,f(\alpha ^{14}){\Big )}=(3,6,15,6,6,3,13,14,3,12,15,2,11,1,0)}" loading="lazy">.<a href="#cite_note-:0-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Stephen B. Wicker, Vijay K. Bhargava: <cite style="font-style:italic">Reed Solomon Codes Applications</cite>. Wiley, 1999, ISBN 0-7803-5391-9 (<a rel="nofollow" class="external text" href="http://ieeexplore.ieee.org/xpl/bkabstractplus.jsp?bkn=5264570">ieeexplore.ieee.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Reed-Solomon-Code&rft.au=Stephen+B.+Wicker%2C+Vijay+K.+Bhargava&rft.btitle=Reed+Solomon+Codes+Applications&rft.date=1999&rft.genre=book&rft.isbn=0780353919&rft.pub=Wiley" style="display:none"> </li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.eccpage.com/">Kodier- und Dekodieralgorithmen für Reed-Solomon- und andere Kodes in C (Robert Morelos-Zaragoza)</a> (en)</li>

<li>James S. Plank: <a rel="nofollow" class="external text" href="http://www.cs.utk.edu/~plank/plank/papers/SPE-9-97.html">A Tutorial on Reed-Solomon Coding for Fault-Tolerance in RAID-like Systems</a>, Software – Practice & Experience, 27(9), September, 1997, S. 995–1012</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

Irving S. Reed, Gustave Solomon: <cite style="font-style:italic">Polynomial codes over certain finite fields</cite>. In: <cite style="font-style:italic">Journal of the Society for Industrial and Applied Mathematics, SIAM J.</cite> Band 8, 1960, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220036-1399%22&key=cql">0036-1399</a>, S. 300–304.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Reed-Solomon-Code&rft.atitle=Polynomial+codes+over+certain+finite+fields&rft.au=Irving+S.+Reed%2C+Gustave+Solomon&rft.date=1960&rft.genre=journal&rft.issn=0036-1399&rft.jtitle=Journal+of+the+Society+for+Industrial+and+Applied+Mathematics%2C+SIAM+J.&rft.pages=300-304&rft.volume=8" style="display:none">

</li>

<li id="cite_note-:0-2">↑ <a href="#cite_ref-:0_2-0">a</a> <a href="#cite_ref-:0_2-1">b</a> Eduard Jorswieck, Anne Wolf, Technische Universität Dresden: <a rel="nofollow" class="external text" href="https://tu-dresden.de/ing/elektrotechnik/ifn/itml/ressourcen/dateien/lehre/vl/codth/A2_RSC.pdf?lang=de">Reed-Solomon-Coder und -Decoder</a>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Benjamin Klopsch: <a rel="nofollow" class="external text" href="https://www.math.uni-duesseldorf.de/~klopsch/mathematics/Manuskripte/cd_rscode.pdf">Audio CDs und Reed-Solomon Codes</a>

</li>

</ol></div><div><div class="zim-footer">

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