GermanWikipedia - /page/entry.mu

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<title>Linearer Code</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Linearer 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">Ein linearer Code ist in der <a href="Kodierungstheorie" title="Kodierungstheorie">Kodierungstheorie</a> ein spezieller <a href="Blockcode" title="Blockcode">Blockcode</a>, bei dem die Codewörter Elemente eines endlichdimensionalen <a href="Vektorraum" title="Vektorraum">Vektorraums</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}^{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a630603c6f2f8f58f5d1fb9ef1b9ad194e258a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.639ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{q}^{n}}" loading="lazy"> über einem <a href="Endlicher_K%C3%B6rper" title="Endlicher Körper">endlichen Körper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}" loading="lazy"> sind. Ein <a href="Code" title="Code">Code</a> ist genau dann linear, wenn er ein <a href="Untervektorraum" title="Untervektorraum">Untervektorraum</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}^{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a630603c6f2f8f58f5d1fb9ef1b9ad194e258a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.639ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{q}^{n}}" loading="lazy"> ist.

Lineare Codes haben den Vorteil, dass Methoden der <a href="Lineare_Algebra" title="Lineare Algebra">Linearen Algebra</a> verwendet werden können. Sie sind somit einfach zu kodieren und dekodieren. Die meisten wichtigen Codes sind linear: <a href="Hamming-Code" title="Hamming-Code">Hamming-Code</a>, <a href="Low-Density-Parity-Check-Code" title="Low-Density-Parity-Check-Code">Low-Density-Parity-Check-Code</a>, <a href="Reed-Muller-Code" title="Reed-Muller-Code">Reed-Muller-Code</a>, <a href="Hadamard-Code" title="Hadamard-Code">Hadamard-Code</a>, alle <a href="Zyklischer_Code" title="Zyklischer Code">zyklischen Codes</a> (damit auch <a href="BCH-Code" title="BCH-Code">BCH</a>, <a href="Reed-Solomon-Code" title="Reed-Solomon-Code">Reed-Solomon-Codes</a>, <a href="Golay-Code" title="Golay-Code">Golay-Codes</a> und Goppa-Codes).

Ist die Vektorraumdimension des linearen Codes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> gleich <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">, so nennt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> einen <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_/eb734a37dd21ce173a46342d1cc64c92/4d9e3e5f03db243c10e64608605d156ae1885952.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.934ex; height:2.843ex;" alt="{\displaystyle [n,k]}" loading="lazy">-Code oder bei einem <a href="Hamming-Abstand" title="Hamming-Abstand">Hamming-Abstand</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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

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</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"> auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n,k,d]}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef7bd63229a2d74455e0037c5d70d39a13184a75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.183ex; height:2.843ex;" alt="{\displaystyle [n,k,d]}" loading="lazy">-Code.

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> ein <a href="Untervektorraum" title="Untervektorraum">Untervektorraum</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> ist, existiert eine <a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1},\dots ,g_{k}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/801375d568ed6292caa4f668207721e8d013a6f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.539ex; height:2.009ex;" alt="{\displaystyle g_{1},\dots ,g_{k}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">. Fasst man diese Basis in einer <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a>

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<annotation encoding="application/x-tex">{\displaystyle G={\begin{pmatrix}g_{1}\\\vdots \\g_{k}\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b76de7b6adb5d9a40f5c5f653814d121345368e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:11.941ex; height:10.509ex;" alt="{\displaystyle G={\begin{pmatrix}g_{1}\\\vdots \\g_{k}\end{pmatrix}}}" loading="lazy"></dd></dl>

zusammen, erhält man eine <a href="Generatormatrix" title="Generatormatrix">Erzeugermatrix</a>. Des Weiteren besitzt der Code eine <a href="Kontrollmatrix" class="mw-redirect" title="Kontrollmatrix">Kontrollmatrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">. Für sie gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Hc^{T}=0}">

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<annotation encoding="application/x-tex">{\displaystyle Hc^{T}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71d672177aeaf13050c0d9f00255ba040a95341f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.721ex; height:2.676ex;" alt="{\displaystyle Hc^{T}=0}" loading="lazy"> genau dann, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> ein Codewort ist. Der <a href="Rang_(Lineare_Algebra)" title="Rang (Lineare Algebra)">Rang</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> ist <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">, der von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> ist <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/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">. Der <a href="Hamming-Abstand#Hamming-Abstand_eines_Codes" title="Hamming-Abstand">Hamming-Abstand</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> ist die minimale Anzahl linear abhängiger Spalten der Kontrollmatrix.

Das <a href="Hamming-Gewicht" title="Hamming-Gewicht">Hamming-Gewicht</a> eines Codewortes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=(c_{1},\dots ,c_{n})}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e229ec125588c0f9fc349ee87550f6a99d5e3dd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.379ex; height:2.843ex;" alt="{\displaystyle c=(c_{1},\dots ,c_{n})}" loading="lazy"> ist gleich der Anzahl der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01acb7953ba52c2aa44264b5d0f8fd223aa178a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}" loading="lazy">, die von Null verschieden sind. Beispielsweise hat das Codewort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (430540)}">

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<annotation encoding="application/x-tex">{\displaystyle (430540)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40f27839581b41264431d699f999c3c6c989eb2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.784ex; height:2.843ex;" alt="{\displaystyle (430540)}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{7}^{6}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{7}^{6}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/624def2eeff9c1351244b8ffb7abcb9d58996f8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.475ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{7}^{6}}" loading="lazy"> das Hamming-Gewicht 4. Der <a href="Hamming-Abstand" title="Hamming-Abstand">Hamming-Abstand</a> des Codes ist gleich dem kleinsten Hamming-Gewicht aller Codewörter außer dem Nullwort.

Permutiert man die einzelnen <a href="Koordinatensystem" title="Koordinatensystem">Koordinaten</a> der Codewörter, erhält man einen sogenannten äquivalenten Code. Damit und mittels <a href="Lineare_Algebra" title="Lineare Algebra">linearer Algebra</a> kann man zu jedem linearen Code einen äquivalenten finden, der eine <a href="Erzeugermatrix" class="mw-redirect" title="Erzeugermatrix">Erzeugermatrix</a> der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=(\mathbb {E} _{k}|P)}">

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<annotation encoding="application/x-tex">{\displaystyle G=(\mathbb {E} _{k}|P)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7862f65cb4c15968bb04cb6f26da98c5acecf8f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.766ex; height:2.843ex;" alt="{\displaystyle G=(\mathbb {E} _{k}|P)}" loading="lazy"> hat. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} _{k}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d49dfdc12800d2112dc4ec8c06042bfedab1c85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.639ex; height:2.509ex;" alt="{\displaystyle \mathbb {E} _{k}}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k\times k)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c495164c369523f0cb052cf42baa835babd8b9ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.072ex; height:2.843ex;" alt="{\displaystyle (k\times k)}" loading="lazy">-<a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a>, und <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/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> ist eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k\times (n-k))}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43eca57b09a46ad956072214a1ac8e5da3acad6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.117ex; height:2.843ex;" alt="{\displaystyle (k\times (n-k))}" loading="lazy">-Matrix. Dann heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> Erzeugermatrix in reduzierter Form. Die ersten <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"> Koordinaten können als Informationssymbole und die letzten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}">

</mstyle>

</mrow>

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

</semantics>

</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"> als Kontrollsymbole interpretiert werden. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> eine Erzeugermatrix in reduzierter Form, kann eine Kontrollmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> sofort gefunden werden: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=(-P^{T}|\mathbb {E} _{n-k})}">

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=(-P^{T}|\mathbb {E} _{n-k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c8f345a2f590633b5495032a0bd895c08ccf7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.541ex; height:3.176ex;" alt="{\displaystyle H=(-P^{T}|\mathbb {E} _{n-k})}" loading="lazy">. Ein linearer Code ist bereits durch seine Erzeugermatrix oder seine Kontrollmatrix bestimmt.

Weil der lineare Code ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</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="Dimension_(Mathematik)" title="Dimension (Mathematik)">dimensionaler</a> <a href="Untervektorraum" title="Untervektorraum">Untervektorraum</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}^{n}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a630603c6f2f8f58f5d1fb9ef1b9ad194e258a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.639ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{q}^{n}}" loading="lazy"> ist, ist er der <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> einer <a href="Lineare_Abbildung" title="Lineare Abbildung">linearen Abbildung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {F} _{q}^{n}\to \mathbb {F} _{q}^{n-k}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {F} _{q}^{n}\to \mathbb {F} _{q}^{n-k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf76028da238bf41e688dda6e015dfe543be6723.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.339ex; height:3.509ex;" alt="{\displaystyle f\colon \mathbb {F} _{q}^{n}\to \mathbb {F} _{q}^{n-k}}" loading="lazy">. Die <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> einer solchen linearen Abbildung heißt Paritätsprüfungsmatrix und wird notiert als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{T}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b11971244315e0ca1923b6375a3d6e8908ebc40c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.493ex; height:2.676ex;" alt="{\displaystyle H^{T}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\in \mathbb {F} _{q}^{(n-k)\times {n}}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\in \mathbb {F} _{q}^{(n-k)\times {n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53bd137beb9a4286879069e79e38b20c44b2898d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.222ex; height:3.509ex;" alt="{\displaystyle H\in \mathbb {F} _{q}^{(n-k)\times {n}}}" loading="lazy">. Der lineare Code ist definiert als <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> dieser linearen Abbildung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C:=\mathrm {ker} (f)=\{c\in \mathbb {F} _{q}^{n}\mid cH^{T}=0\}}">

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C:=\mathrm {ker} (f)=\{c\in \mathbb {F} _{q}^{n}\mid cH^{T}=0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03a35e05df7e4816f7c8fdf6b13b07470cca6764.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:34.378ex; height:3.509ex;" alt="{\displaystyle C:=\mathrm {ker} (f)=\{c\in \mathbb {F} _{q}^{n}\mid cH^{T}=0\}}" loading="lazy">.

Die <a href="Informationsrate" title="Informationsrate">Informationsrate</a> des linearen Codes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\tfrac {k}{n}}\cdot \log _{2}(q)}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R={\tfrac {k}{n}}\cdot \log _{2}(q)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b15f5bae17bf98c639107d7bd0a9d10b343f3b88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.269ex; height:3.509ex;" alt="{\displaystyle R={\tfrac {k}{n}}\cdot \log _{2}(q)}" loading="lazy">.<a href="#cite_note-1">[1]</a>

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

Der binäre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [7,4,3]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [7,4,3]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c7cffdce4b8a86c00d513d0aec1c5e32a53f621.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle [7,4,3]}" loading="lazy">-<a href="Hamming-Code" title="Hamming-Code">Hamming-Code</a> besitzt folgende Erzeugermatrix in reduzierter Form sowie die dazugehörige Kontrollmatrix:

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=(\mathbb {E} _{k}|P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7862f65cb4c15968bb04cb6f26da98c5acecf8f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.766ex; height:2.843ex;" alt="{\displaystyle G=(\mathbb {E} _{k}|P)}" loading="lazy">

</td>

<td>

</td>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=(-P^{T}|\mathbb {E} _{n-k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c8f345a2f590633b5495032a0bd895c08ccf7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.541ex; height:3.176ex;" alt="{\displaystyle H=(-P^{T}|\mathbb {E} _{n-k})}" loading="lazy">

</td></tr>

<tr>

<mrow>

<mtr>

<mtd>

</mstyle>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

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<mtd>

</mtd>

<mtd>

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<mtd>

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<mtd>

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<mtr>

<mtd>

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<mtd>

</mstyle>

</mrow>

</mtd>

<mtd>

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<mtd>

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<mtd>

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<mtd>

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<mtr>

<mtd>

</mtd>

<mtd>

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<mtd>

</mrow>

</mstyle>

</mtd>

<mtd>

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<mtd>

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<mtd>

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<mtr>

<mtd>

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<mtd>

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<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G={\begin{pmatrix}{\color {Brown}1}&0&0&0&1&1&0\\0&{\color {Brown}1}&0&0&1&0&1\\0&0&\color {Brown}{1}&0&1&1&1\\0&0&0&{\color {Brown}1}&0&1&1\\\end{pmatrix}}\qquad }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80266085f3fa92b96664a541c425b36ba9a57f1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:36.461ex; height:12.509ex;" alt="{\displaystyle G={\begin{pmatrix}{\color {Brown}1}&0&0&0&1&1&0\\0&{\color {Brown}1}&0&0&1&0&1\\0&0&\color {Brown}{1}&0&1&1&1\\0&0&0&{\color {Brown}1}&0&1&1\\\end{pmatrix}}\qquad }" loading="lazy">

</td>

<td>

</td>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H={\begin{pmatrix}1&1&1&0&{\color {Brown}1}&0&0\\1&0&1&1&0&{\color {Brown}1}&0\\0&1&1&1&0&0&{\color {Brown}1}\\\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2922ce3ae8d0bc83966df9458c6ee0c861ca44d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:32.053ex; height:9.176ex;" alt="{\displaystyle H={\begin{pmatrix}1&1&1&0&{\color {Brown}1}&0&0\\1&0&1&1&0&{\color {Brown}1}&0\\0&1&1&1&0&0&{\color {Brown}1}\\\end{pmatrix}}}" loading="lazy">

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

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

Ein Wort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(x_{1},\dots ,x_{k})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd77c98584c3731108ea1df6ad29b354025fd482.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.218ex; height:2.843ex;" alt="{\displaystyle x=(x_{1},\dots ,x_{k})}" loading="lazy"> aus dem Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}^{k}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c424754306a34bb549057005ea648c90f28dee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.509ex; height:3.509ex;" alt="{\displaystyle \mathbb {F} _{q}^{k}}" loading="lazy"> wird kodiert, indem das Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xG}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60651f5c3dfdebf6aae78e6664264ac28efd8383.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.156ex; height:2.176ex;" alt="{\displaystyle xG}" loading="lazy"> gebildet wird. Die Kodierung des Wortes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,1,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,1,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a2a04e62eebb2bfc5689c61fe861575b7dd542b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.561ex; height:2.843ex;" alt="{\displaystyle (0,0,1,1)}" loading="lazy"> mit dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [7,4,3]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [7,4,3]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c7cffdce4b8a86c00d513d0aec1c5e32a53f621.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle [7,4,3]}" loading="lazy">-<a href="Hamming-Code" title="Hamming-Code">Hamming-Code</a> veranschaulicht beispielsweise die folgende Rechnung.

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

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<mtd>

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<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

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<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

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<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

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<mtd>

</mtd>

<mtd>

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</mtr>

</mtable>

</mrow>

</mtd>

<mtd>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x\cdot G&=c\\{\begin{pmatrix}0,0,1,1\end{pmatrix}}\cdot {\begin{pmatrix}1&0&0&0&1&1&0\\0&1&0&0&1&0&1\\0&0&1&0&1&1&1\\0&0&0&1&0&1&1\\\end{pmatrix}}&={\begin{pmatrix}0,0,1,1,1,0,0\end{pmatrix}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f48e9d09b4ab463f7fa7bf175908ab2a62c23a86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:59.634ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}x\cdot G&=c\\{\begin{pmatrix}0,0,1,1\end{pmatrix}}\cdot {\begin{pmatrix}1&0&0&0&1&1&0\\0&1&0&0&1&0&1\\0&0&1&0&1&1&1\\0&0&0&1&0&1&1\\\end{pmatrix}}&={\begin{pmatrix}0,0,1,1,1,0,0\end{pmatrix}}\end{aligned}}}" loading="lazy"></dd></dl>

Da hier die Addition in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fde97f1e971e76227cd0aac645b7b0901d7b668d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{2}}" loading="lazy"> erfolgt, gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a94a15bb0333c813ac2ed0d697e5e57092cbbc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+1=0}" loading="lazy">

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

Mit Dekodierung bezeichnet man das Zuordnen eines empfangenen, möglicherweise fehlerhaften Eingabevektors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> zu einem Codevektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f295e32663780aa8800e3106dba9d6e04e57cdc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.692ex; height:2.509ex;" alt="{\displaystyle c'}" loading="lazy">. Die Dekodierung ist nicht die <a href="Umkehrfunktion" title="Umkehrfunktion">Umkehrfunktion</a> der Kodierung, die einem Codevektor wieder einen Vektor aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}^{k}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c424754306a34bb549057005ea648c90f28dee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.509ex; height:3.509ex;" alt="{\displaystyle \mathbb {F} _{q}^{k}}" loading="lazy"> zuordnet.

Als Dekodierungsmethode wird in der Kodierungstheorie meistens die Methode der größten Wahrscheinlichkeit (englisch: maximum likelihood decoding) verwendet. Dabei wird ein empfangener Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> zu dem Codevektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f295e32663780aa8800e3106dba9d6e04e57cdc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.692ex; height:2.509ex;" alt="{\displaystyle c'}" loading="lazy"> dekodiert, der mit der größten Wahrscheinlichkeit zum tatsächlich versandten Codevektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> identisch ist. Häufig wird der Vektor, bei dem die wenigsten Stellen (Fehler) korrigiert werden müssen, als der Wahrscheinlichste angenommen. Mathematisch gesprochen heißt das, man sucht den Codevektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f295e32663780aa8800e3106dba9d6e04e57cdc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.692ex; height:2.509ex;" alt="{\displaystyle c'}" loading="lazy"> mit dem geringsten Hamming-Abstand zum empfangenen Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. Dieser Fall wird auch als Methode des nächstgelegenen Nachbarn (englisch: nearest neighbor decoding) bezeichnet. Durch Kenntnis der Art der gesendeten Daten oder des verwendeten <a href="Kanal_(Informationstheorie)" title="Kanal (Informationstheorie)">Kanals</a> können gegebenenfalls andere Informationen verwendet werden, um die Wahrscheinlichkeit für bestimmte Codevektoren zu bestimmen.

Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> der tatsächlich versendete (Code-)Vektor und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> der empfangene Vektor. Die Dekodierung sucht aus allen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31246d3db82f98a975bd9d9a5ea525679f89ff6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.298ex; height:2.676ex;" alt="{\displaystyle q^{n}}" loading="lazy"> Codevektoren den oder die Codevektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f295e32663780aa8800e3106dba9d6e04e57cdc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.692ex; height:2.509ex;" alt="{\displaystyle c'}" loading="lazy">, die mit der größten Wahrscheinlichkeit versendet wurden.

<msup>

</msup>

<msub>

<mi>argmax</mi>

</mrow>

</msub>

<mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c'=\operatorname {argmax} _{y\in C}\left\{\operatorname {P} \left[y=c\right]\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ff42f0d1bd425b70e4504d9bb5c45bf12b44747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.209ex; height:3.343ex;" alt="{\displaystyle c'=\operatorname {argmax} _{y\in C}\left\{\operatorname {P} \left[y=c\right]\right\}}" loading="lazy"></dd></dl>

Bei der Nearest-Neighbor-Dekodierung also:

<msup>

</msup>

<msub>

<mi>argmin</mi>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c'=\operatorname {argmin} _{y\in C}\left\{wt(x-y)\right\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64ff10646e9f017364035f347de7db530c79b606.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.26ex; height:3.343ex;" alt="{\displaystyle c'=\operatorname {argmin} _{y\in C}\left\{wt(x-y)\right\}}" loading="lazy"></dd></dl>

Man sollte dabei beachten, dass diese Zuordnung bei den meisten Codes nicht für alle Fehlervektoren eindeutig ist. Es gibt dann einige Fehlervektoren, die nicht zugeordnet werden können, da sie mehr als einen nächstgelegenen Nachbarn haben. Ist für jeden Fehlervektor eine eindeutige Dekodierung möglich, so heißt der Code <a href="Perfekter_Code" title="Perfekter Code">perfekt</a>.

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

Eine effizientere Methode für die Dekodierung stellt die sogenannte Syndrom-Dekodierung dar. Das Syndrom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f738a5f93f694ded413a8f8da71081d0fdff4501.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.009ex;" alt="{\displaystyle s_{x}}" loading="lazy"> eines Vektors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> erhält man durch Multiplikation der Kontrollmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{T}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d695dd7f6a572d2c6941efef9dc2dd15bff6ee00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.719ex; height:2.676ex;" alt="{\displaystyle x^{T}}" loading="lazy">.

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{x}^{T}=Hx^{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63e93da2218c1d9566f97224a4a13882079abd1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.361ex; height:3.009ex;" alt="{\displaystyle s_{x}^{T}=Hx^{T}}" loading="lazy"></dd></dl>

Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=x-c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8060215e26644f5d5454310cc855a34dfbf15967.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.359ex; height:2.176ex;" alt="{\displaystyle e=x-c}" loading="lazy"> der Fehlervektor von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> sind genau die Koordinaten ungleich Null, bei denen während der Übertragung Fehler aufgetreten sind.

Für das Syndrom von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> gilt wegen der Linearität des Codes:

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{x}^{T}=Hx^{T}=Hc^{T}+He^{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bb80dd8386308ae8020877e5cda3b1ccd687bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.296ex; height:3.009ex;" alt="{\displaystyle s_{x}^{T}=Hx^{T}=Hc^{T}+He^{T}}" loading="lazy"></dd></dl>

Da das Syndrom von fehlerfreien Codevektoren immer Null ist, folgt:

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{x}^{T}=He^{T}=s_{e}^{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4638d09e1cf8225c25af3084482bead160f8b46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.693ex; height:3.009ex;" alt="{\displaystyle s_{x}^{T}=He^{T}=s_{e}^{T}}" loading="lazy"></dd></dl>

Alle (fehlerhaften) Wörter mit dem gleichen Fehlervektor sind im gleichen <a href="Affiner_Raum" title="Affiner Raum">affinen Unterraum</a>, das heißt für solche Wörter ist das Syndrom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{e}^{T}=He^{T}}">

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{e}^{T}=He^{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e5e4d96c2ba35c9714ebb2b896e182c8d6da2e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.115ex; height:3.009ex;" alt="{\displaystyle s_{e}^{T}=He^{T}}" loading="lazy"> konstant.

Alle Vektoren, die aus einem beliebigen, festen Vektor durch Subtraktion eines beliebigen Codevektors hervorgegangen sind, bilden eine <a href="Nebenklasse_(Mathematik)" class="mw-redirect" title="Nebenklasse (Mathematik)">Nebenklasse</a> der <a href="Untergruppe" title="Untergruppe">Untergruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}^{n}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a630603c6f2f8f58f5d1fb9ef1b9ad194e258a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.639ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{q}^{n}}" loading="lazy">. Der Vektor mit minimalem Gewicht in dieser Klasse heißt Führer der Nebenklasse (englisch: coset leader). Daher ist auch der Begriff „coset leader decoding“ verbreitet.

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f295e32663780aa8800e3106dba9d6e04e57cdc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.692ex; height:2.509ex;" alt="{\displaystyle c'}" loading="lazy"> zu dekodieren, sucht man also den Fehlervektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06c198f6710d781baaf94653df305a4881380033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.768ex; height:2.509ex;" alt="{\displaystyle e'}" loading="lazy">, dessen Syndrom identisch zum Syndrom von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> ist und dessen <a href="Hamming-Gewicht" title="Hamming-Gewicht">Hamming-Gewicht</a> minimal ist. Mit diesem Fehlervektor berechnet man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c'=x-e'}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c'=x-e'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cf4a2ccf22ce9c19ea4abaffecfdb29a3fda8b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.728ex; height:2.676ex;" alt="{\displaystyle c'=x-e'}" loading="lazy"> den nächstgelegenen Codevektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f295e32663780aa8800e3106dba9d6e04e57cdc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.692ex; height:2.509ex;" alt="{\displaystyle c'}" loading="lazy">.

Man kann also eine Tabelle mit bis zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{n-k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cf44f5c6a300293184c889ce4242976b00552c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.433ex; height:3.009ex;" alt="{\displaystyle q^{n-k}}" loading="lazy"> Zeilen aufstellen, die für jedes Syndrom eines empfangenen Vektors den entsprechenden Fehlervektor mit minimalem Hamming-Gewicht beinhaltet. Ist das Syndrom gleich 0, dann muss nichts korrigiert werden ansonsten beschränkt sich Dekodierung auf das Nachschlagen des Fehlervektors in dieser Tabelle und Korrigieren der so detektierten Fehler.

Anders interpretiert sind die Nebenklassen gerade die <a href="%C3%84quivalenzklasse" class="mw-redirect" title="Äquivalenzklasse">Äquivalenzklassen</a> der <a href="%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\equiv y\Leftrightarrow s_{x}^{T}=s_{y}^{T}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\equiv y\Leftrightarrow s_{x}^{T}=s_{y}^{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d948c793016d02672137d794a37c133cb723d3eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.256ex; height:3.176ex;" alt="{\displaystyle x\equiv y\Leftrightarrow s_{x}^{T}=s_{y}^{T}}" loading="lazy"> Die Führer sind gerade Vertreter der Äquivalenzklassen, man wählt einen mit minimalem Hamming-Gewicht. <a href="Perfekter_Code" title="Perfekter Code">Perfekte Codes</a> zeichnen sich dadurch aus, dass die Führer eindeutig festgelegt sind.

Die Dekodierung von linearen Codes ist im Allgemeinen <a href="NP-Vollst%C3%A4ndigkeit" title="NP-Vollständigkeit">NP-vollständig</a>, das heißt, es sind keine Algorithmen mit polynomieller Laufzeit bekannt.<a href="#cite_note-2">[2]</a>

Die bekannten linearen Codes, beispielsweise Hamming-Codes, zeichnen sich dadurch aus, dass für sie effiziente Dekodierungs-Algorithmen bekannt sind.

Die Komplexität der linearen Dekodierung ist Grundlage für das <a href="McEliece-Kryptosystem" title="McEliece-Kryptosystem">McEliece-Kryptosystem</a>, das als sicher gilt, aber aufgrund seiner vergleichsweise langen Schlüssel bisher selten eingesetzt wird.

<div class="mw-heading mw-heading4"><h4 id="Beispiel_2">Beispiel</h4></div>

Will man den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [7,4,3]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [7,4,3]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c7cffdce4b8a86c00d513d0aec1c5e32a53f621.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle [7,4,3]}" loading="lazy">-Hamming-Code (von oben) dekodieren, trifft man zuerst die Annahme, dass nur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">-<a href="Bit" title="Bit">Bit</a> Fehler auftreten. Die möglichen Fehlervektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> sind dann

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,0,0,0,0,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36535fea822e768773af41ca6530d421e3a0c1e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (1,0,0,0,0,0,0)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,1,0,0,0,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0f0cfeacb98db100932bf8eeaafa5ddff1d32b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,1,0,0,0,0,0)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,1,0,0,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c76de279c8afa63dc9db7e8f042ab0db0ff9b379.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,0,1,0,0,0,0)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,0,1,0,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a89523a716788f96f92dcdb21156c799454b8433.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,0,0,1,0,0,0)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,0,0,1,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dd07919f1eb73044f3a5ca3fb008dcff9d5eb8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,0,0,0,1,0,0)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,0,0,0,1,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a999cb5fe7243cff1131291ea2d943a35ac7b859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,0,0,0,0,1,0)}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,0,0,0,0,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db31c6c634c7e9289c3efabf46a17f1044b340d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,0,0,0,0,0,1)}" loading="lazy"></dd></dl>

Für jeden dieser Fehlervektoren wird nun das Syndrom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{T}=He^{T}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s^{T}=He^{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c2a5a194e4ec08187ae3088eec2157c631b9982.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.115ex; height:2.676ex;" alt="{\displaystyle s^{T}=He^{T}}" loading="lazy"> berechnet.

Damit ergibt sich

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy">

</th>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,0,0,0,0,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36535fea822e768773af41ca6530d421e3a0c1e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (1,0,0,0,0,0,0)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,1,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5568de79a98cdcb48821cd81b670c095fa4c8b6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (1,1,0)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,1,0,0,0,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0f0cfeacb98db100932bf8eeaafa5ddff1d32b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,1,0,0,0,0,0)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,0,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e62ee87058de1e6f24731a117229cf7020b14ab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (1,0,1)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,1,0,0,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c76de279c8afa63dc9db7e8f042ab0db0ff9b379.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,0,1,0,0,0,0)}" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efeac0c817a90342bab7878eb40cb33dd6facbbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (1,1,1)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,0,1,0,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a89523a716788f96f92dcdb21156c799454b8433.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,0,0,1,0,0,0)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,1,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57c2df6cd05d09c31ff612e7b003514d067dc03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,1,1)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,0,0,1,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dd07919f1eb73044f3a5ca3fb008dcff9d5eb8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,0,0,0,1,0,0)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ea6e8ece35797224448db97fa0ea17544a7f756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (1,0,0)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,0,0,0,1,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a999cb5fe7243cff1131291ea2d943a35ac7b859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,0,0,0,0,1,0)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,1,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e28bfe8a44d06939011e3ba2159e894c0f22f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,1,0)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,0,0,0,0,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db31c6c634c7e9289c3efabf46a17f1044b340d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle (0,0,0,0,0,0,1)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25b35543ef5f96df2e9aea6cced9bdd57dab7e1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,0,1)}" loading="lazy">

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

Wird dann das fehlerhafte Wort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(0,1,1,1,1,0,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=(0,1,1,1,1,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0772f67f3ee26daceb8c709f1746ca961ec4f2a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.578ex; height:2.843ex;" alt="{\displaystyle x=(0,1,1,1,1,0,0)}" loading="lazy"> empfangen, ergibt dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{T}=Hx^{T}=(1,0,1)^{T}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s^{T}=Hx^{T}=(1,0,1)^{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07b9c813c951998b9fc2558cc005698c9006b22f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.213ex; height:3.176ex;" alt="{\displaystyle s^{T}=Hx^{T}=(1,0,1)^{T}}" loading="lazy">. Damit ergibt sich der Fehlervektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=e((1,0,1))=(0,1,0,0,0,0,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e=e((1,0,1))=(0,1,0,0,0,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/924fb41cdf43f8b9d15e379d5e6da7df790f0d8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.688ex; height:2.843ex;" alt="{\displaystyle e=e((1,0,1))=(0,1,0,0,0,0,0)}" loading="lazy">, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> wird somit nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x)=(0,1,1,1,1,0,0)-(0,1,0,0,0,0,0)=(0,0,1,1,1,0,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c(x)=(0,1,1,1,1,0,0)-(0,1,0,0,0,0,0)=(0,0,1,1,1,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f15e379e4d9fdd44680f9634bd5e4cff70a8510.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.634ex; height:2.843ex;" alt="{\displaystyle c(x)=(0,1,1,1,1,0,0)-(0,1,0,0,0,0,0)=(0,0,1,1,1,0,0)}" loading="lazy"> dekodiert. Das Klartextwort ist dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0,1,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,1,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a2a04e62eebb2bfc5689c61fe861575b7dd542b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.561ex; height:2.843ex;" alt="{\displaystyle (0,0,1,1)}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Beispiel_mit_unvollständiger_Dekodierung">Beispiel mit unvollständiger Dekodierung</h4></div>

Gegeben sei der ternäre (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=3}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d859ff2e5023ed1f714ccace69e88ab993a5f43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=3}" loading="lazy">) Wiederholungscode der Länge 3:

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G={\begin{pmatrix}1&1&1\end{pmatrix}}\qquad }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f74652187faecbd8776930d1ce92bdee905061b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.264ex; height:2.843ex;" alt="{\displaystyle G={\begin{pmatrix}1&1&1\end{pmatrix}}\qquad }" loading="lazy">

</td>

<td>

</td>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H={\begin{pmatrix}2&1&0\\2&0&1\\\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc9264f209b08e79ff228cd4498c54a4635ae4fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.467ex; height:6.176ex;" alt="{\displaystyle H={\begin{pmatrix}2&1&0\\2&0&1\\\end{pmatrix}}}" loading="lazy">

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

Jeweils zwei Spalten von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> sind linear unabhängig, alle drei dagegen linear abhängig. Der minimale Hamming-Abstand des Codes berechnet sich als minimale Anzahl linear abhängiger Spalten in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> also zu 3. Es kann damit höchstens ein Zeichenfehler korrigiert werden. Die Syndromtabelle sieht folgendermaßen aus:

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy">

</th>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ea6e8ece35797224448db97fa0ea17544a7f756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (1,0,0)}" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94d0ca163bd6ff9c2757666390d7a4c0ee7273b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (2,2)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (2,0,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a55ae623f6199e05185701003adc933e0243db81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (2,0,0)}" loading="lazy">

</td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c2a42feb07f4139bf871ae6856b11d4567bea23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,1)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,1,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e28bfe8a44d06939011e3ba2159e894c0f22f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,1,0)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b53cc1773694affcc1d4d6c2c778d43156a1206.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,0)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,2,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a6e862158ad77affea1a3661742da745a421391.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,2,0)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (2,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03972a1096453eb69d2337a01165887760403a8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (2,0)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25b35543ef5f96df2e9aea6cced9bdd57dab7e1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,0,1)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,0,2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/803a56c098d9c2b7686dbaeaa46d9f3bea72fd08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle (0,0,2)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32ba2704e55769d5190aaa9493a088b7df8e3996.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,2)}" loading="lazy">

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

Durch Ausnützen von Linearitäten könnte die Anzahl der Zeilen halbiert werden, man muss dann aber testen, ob ein linear abhängiges Syndrom in der Tabelle ist.

Man betrachte nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(1,1,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=(1,1,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3012da1abe571cc3d3b2249ff74af29a3a4e158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.793ex; height:2.843ex;" alt="{\displaystyle x=(1,1,0)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=(1,2,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=(1,2,0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb8a6fc1a88e09d1d8f4ddce3a240b3f6049d5ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.618ex; height:2.843ex;" alt="{\displaystyle y=(1,2,0)}" loading="lazy">. Bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> ist das Syndrom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Hx^{t}=(0,2)^{t}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Hx^{t}=(0,2)^{t}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd5ea9582d0747dd3c6c3ecc38aa183b4924ed1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.312ex; height:3.009ex;" alt="{\displaystyle Hx^{t}=(0,2)^{t}}" loading="lazy">, die Korrektur lautet: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,1,0)-(0,0,2)=(1,1,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,1,0)-(0,0,2)=(1,1,1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/091c03c41599c52d549fe21deb380b4ae68f8c81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.032ex; height:2.843ex;" alt="{\displaystyle (1,1,0)-(0,0,2)=(1,1,1)}" loading="lazy">. Die Berechnung des Syndroms von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> ergibt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Hy^{t}=(1,2)^{t}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Hy^{t}=(1,2)^{t}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/310dccced669731c699e2921ca211c2acbae3cc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.143ex; height:3.009ex;" alt="{\displaystyle Hy^{t}=(1,2)^{t}}" loading="lazy">. Dieser Wert ist nicht in der Syndromtabelle enthalten, das Wort kann also nicht korrigiert werden.

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

Die Kodierung und Dekodierung ist, so wie sie oben beschrieben ist, relativ aufwendig. Bei der Kodierung muss die Erzeugermatrix im Speicher gehalten werden, was bei Systemen mit begrenzten Ressourcen (zum Beispiel mobile Endgeräte oder Weltraumsonden) problematisch ist. Bei der Dekodierung wird eine – je nach Korrekturrate – große Tabelle benötigt; der Speicherverbrauch ist dementsprechend groß. Aus diesem Grund werden in der Regel zusätzliche Eigenschaften der Codes benutzt, um diese effizient zu kodieren und dekodieren. Binäre <a href="Zyklischer_Code" title="Zyklischer Code">zyklische Codes</a> lassen sich beispielsweise sehr einfach mittels <a href="Schieberegister" title="Schieberegister">Schieberegister</a> und <a href="Exklusiv-Oder-Gatter" title="Exklusiv-Oder-Gatter">Exklusiv-Oder-Gatter</a> realisieren.

<div class="mw-heading mw-heading2"><h2 id="Dualer_Code">Dualer Code</h2></div>

Zu jedem (linearen) Code <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> gibt es einen dualen Code (oder auch Dualcode) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\perp }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2329fab4b0d3cf8941ef7197caf13ef920270b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.309ex; height:2.676ex;" alt="{\displaystyle C^{\perp }}" loading="lazy">, der selbst ein linearer Code ist. Die Codewörter des dualen Codes sind alle Wörter aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}^{n}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a630603c6f2f8f58f5d1fb9ef1b9ad194e258a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.639ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{q}^{n}}" loading="lazy">, die zu den Codewörtern aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> dual sind:

Man definiert hierzu ein inneres Produkt:

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ;\cdot \rangle \,:\;\mathbb {F} _{q}^{n}\times \mathbb {F} _{q}^{n}\to \mathbb {F} _{q}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a6db1c353b99839365c99391befd906683e78f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.247ex; height:3.176ex;" alt="{\displaystyle \langle \cdot ;\cdot \rangle \,:\;\mathbb {F} _{q}^{n}\times \mathbb {F} _{q}^{n}\to \mathbb {F} _{q}}" loading="lazy"></dd></dl>

das die Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(x_{1},x_{2},\dots ,x_{n}),y=(y_{1},y_{2},\dots ,y_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=(x_{1},x_{2},\dots ,x_{n}),y=(y_{1},y_{2},\dots ,y_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7210dba63b763b76f55b7fed334f297a29135cd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.82ex; height:2.843ex;" alt="{\displaystyle x=(x_{1},x_{2},\dots ,x_{n}),y=(y_{1},y_{2},\dots ,y_{n})}" loading="lazy"> folgendermaßen abbildet:

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle x;y\rangle =\sum _{i=1}^{n}x_{i}y_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c25856e5448f2cd2d168e028bca1c45adec8ee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.237ex; height:6.843ex;" alt="{\displaystyle \langle x;y\rangle =\sum _{i=1}^{n}x_{i}y_{i}}" loading="lazy"></dd></dl>

Trotz der ähnlichen Definition handelt es sich hierbei nicht um ein <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a>, da diese <a href="Bilinearform" title="Bilinearform">Bilinearform</a> nicht <a href="Positiv_definit" class="mw-redirect" title="Positiv definit">positiv definit</a> ist. Es gibt nämlich aufgrund der Eigenschaften von Endlichen Körpern meistens Vektoren, die ungleich dem <a href="Nullvektor" title="Nullvektor">Nullvektor</a> sind und bei denen das innere Produkt 0 ergibt. Man denke beispielsweise an den binären Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,1)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c2a42feb07f4139bf871ae6856b11d4567bea23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,1)}" loading="lazy">.

Mit Hilfe dieser Definition ergibt sich der Duale Code als:

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<annotation encoding="application/x-tex">{\displaystyle C^{\perp }=\{x\in \mathbb {F} _{q}^{n}\mid \langle x,c\rangle =0\;\mathrm {f{\ddot {u}}r\;alle} \;c\in C\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c01c58c6af976718af0c9bd65225e6aa85c7e225.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.026ex; height:3.509ex;" alt="{\displaystyle C^{\perp }=\{x\in \mathbb {F} _{q}^{n}\mid \langle x,c\rangle =0\;\mathrm {f{\ddot {u}}r\;alle} \;c\in C\}}" loading="lazy"></dd></dl>

Eine Erzeugermatrix des dualen Codes ist eine Kontrollmatrix des Ursprungscodes und umgekehrt.

Der duale Code spielt bei der Analyse der Eigenschaften von Codes eine wichtige Rolle.

Ein Spezialfall sind die sogenannten selbstdualen Codes. Dies sind Codes, die mit ihrem Dualcode identisch sind. Aus Dimensiongründen haben diese immer die Dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=n/2}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e121b1ddb69638f4126c92f975fbf1c36d78087c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.029ex; height:2.843ex;" alt="{\displaystyle k=n/2}" loading="lazy">.

Das wichtigste Beispiel für einen selbstdualen Code ist der erweiterte Hamming-Code, bei dem der binäre [7,4,3]-Hamming-Code um ein <a href="Parit%C3%A4tsbit" title="Paritätsbit">Paritätsbit</a> auf gerade Parität erweitert wird:

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<annotation encoding="application/x-tex">{\displaystyle G=H={\begin{pmatrix}1&0&0&0&1&1&0&1\\0&1&0&0&1&0&1&1\\0&0&1&0&1&1&1&0\\0&0&0&1&0&1&1&1\\\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c6d649a1e0e27a1d60909b8acf119ad97176efc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:40.463ex; height:12.509ex;" alt="{\displaystyle G=H={\begin{pmatrix}1&0&0&0&1&1&0&1\\0&1&0&0&1&0&1&1\\0&0&1&0&1&1&1&0\\0&0&0&1&0&1&1&1\\\end{pmatrix}}}" loading="lazy"></dd></dl>

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

<ul><li><a href="Werner_L%C3%BCtkebohmert" title="Werner Lütkebohmert">Werner Lütkebohmert</a>: Codierungstheorie. Algebraisch-geometrische Grundlagen und Algorithmen. Vieweg Verlag, Braunschweig u. a. 2003, ISBN 3-528-03197-2 (Vieweg-Studium – Aufbaukurs Mathematik).</li>

<li><a href="Jacobus_van_Lint" title="Jacobus van Lint">J. H. van Lint</a>: Introduction to Coding Theory. 3. revised and expanded edition. Springer Verlag, Heidelberg u. a. 1999, ISBN 3-540-64133-5 (Graduate texts in mathematics 86).</li>

<li><a href="F._Jessie_MacWilliams" title="F. Jessie MacWilliams">Florence J. MacWilliams</a>, <a href="Neil_Sloane" title="Neil Sloane">Neil J. Sloane</a>: The Theory of Error-Correcting Codes. 2. Printing. North-Holland publishing company, Amsterdam 1978, ISBN 0-444-85009-0 (North Holland mathematical library 16).</li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Technische Universität Berlin: <a rel="nofollow" class="external text" href="https://www.commit.tu-berlin.de/fileadmin/fg302/CC-linear-codes.pdf">Linear Codes</a>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> E. R. Berlekamp, R. J. McEliece, H. C. A. von Tilburg: <cite style="font-style:italic">On the inherent intractability of certain coding problems</cite>. In: <cite style="font-style:italic">IEEE Transactions on Information Theory 24</cite>. 1978.<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:Linearer+Code&rft.atitle=On+the+inherent+intractability+of+certain+coding+problems&rft.au=E.+R.+Berlekamp%2C+R.+J.+McEliece%2C+H.+C.+A.+von+Tilburg&rft.btitle=IEEE+Transactions+on+Information+Theory+24&rft.date=1978&rft.genre=book" style="display:none">

</li>

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