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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Systematischer Code</span></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"><p>Ein <b>systematischer Code</b> ist ein Code, der jedem Nachrichtenwort der Länge <i>k</i> ein <a href="Code" title="Code">Codewort</a> der Länge <i>n</i> zuordnet, wobei das Nachrichtenwort explizit Teil des zugeordneten Codewortes ist.
</p><p>Als Beispiel gelten <a href="Parit%C3%A4tsbit" title="Paritätsbit">Paritätsprüfungen</a>, bei denen dem Nachrichtenwort ein oder mehrere Prüfbits angehängt werden.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Ein systematischer [<i>n,k,d</i>]-Code kann zum Beispiel so aussehen: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\{(000),(011),(101),(110)\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mo stretchy="false">(</mo>
<mn>000</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo stretchy="false">(</mo>
<mn>011</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo stretchy="false">(</mo>
<mn>101</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo stretchy="false">(</mo>
<mn>110</mn>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C=\{(000),(011),(101),(110)\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6a1ddf287fb3e326b1a92b907bf59ea7947950e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.478ex; height:2.843ex;" alt="{\displaystyle C=\{(000),(011),(101),(110)\}}" loading="lazy"></span>. Dieser Code ließe sich durch folgende Funktion beschreiben: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\{0,1\}^{2}\to \{0,1\}^{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<msup>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<msup>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:\{0,1\}^{2}\to \{0,1\}^{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3edb8ddaa21e5b3a56d5926b205d375450e62b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.306ex; height:3.176ex;" alt="{\displaystyle f:\{0,1\}^{2}\to \{0,1\}^{3}}" loading="lazy"></span>. Und im Einzelnen hätten wir:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,(00)\mapsto (000)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mn>00</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">(</mo>
<mn>000</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,(00)\mapsto (000)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c1d5deca6126950fcae41e2209934e2040e6419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.432ex; height:2.843ex;" alt="{\displaystyle \,(00)\mapsto (000)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,(01)\mapsto (011)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mn>01</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">(</mo>
<mn>011</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,(01)\mapsto (011)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14dc29dcf5ac9eb9dc0e09d616d54bfb91b4346f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.432ex; height:2.843ex;" alt="{\displaystyle \,(01)\mapsto (011)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,(10)\mapsto (101)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">(</mo>
<mn>101</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,(10)\mapsto (101)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c0b3a9a18bc782cccfc03385efdb9a43db43db5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.432ex; height:2.843ex;" alt="{\displaystyle \,(10)\mapsto (101)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,(11)\mapsto (110)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mn>11</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">(</mo>
<mn>110</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \,(11)\mapsto (110)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef362498ba074e37f70bc3cb9437e2c589f26d1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.432ex; height:2.843ex;" alt="{\displaystyle \,(11)\mapsto (110)}" loading="lazy"></span>.</dd></dl>
<p>Nun haben wir einen in 2 Stellen systematischen Code der Länge 3 mit dem Minimalabstand 2 (<a href="Hamming-Abstand" title="Hamming-Abstand">Hamming-Abstand</a>); in Kurzschreibweise also einen [3,2,2]-Code.
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