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Mathematik-Online lexicon:

Linear Code

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Find for the linear Code $ \mbox{$C\subseteq\mathbb{F}_2^4$}$ with generator matrix

$ \mbox{$\displaystyle
G := \left(\begin{matrix}1&0&1&1\\  0&1&1&1\end{matrix}\right)
its minimal distance, information rate and check matrix.

The code words are $ \mbox{$0000$}$, $ \mbox{$1011$}$, $ \mbox{$0111$}$, $ \mbox{$1100$}$. The minimal distance is $ \mbox{$d(C) = 2$}$, the information rate is $ \mbox{$r(C) = \frac{1}{2}$}$ and as a check matrix we get

$ \mbox{$\displaystyle
H=\left(\begin{matrix}1&1\\  1&1\\  1&0\\  0&1\end{matrix}\right).
Therefore its a $ \mbox{$[4,2,2]$}$-code.

Note, that the minimal distance is smaller than $ \mbox{$d(x,0)$}$ for all rows $ \mbox{$x$}$ of the generator matrix.



  automatisch erstellt am 6.  7. 2005