Mathematik-Online lexicon: Linear Code

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	Mathematik-Online lexicon:
	Linear Code

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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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automatisch erstellt am 6. 7. 2005