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Linear Code

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

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.

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see also:

  automatisch erstellt am 6.  7. 2005