# Linear Codes : Error detection

#### mathmari

##### Well-known member
MHB Site Helper
Hey!! The code words of a linear code $C$ have the length $n=5$.

Writing the code words into a matrix to get the linear independent ones, we get the following:
\begin{equation*}\begin{pmatrix}0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0\end{pmatrix}\rightarrow \ldots \rightarrow \begin{pmatrix}1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{pmatrix}\end{equation*}

So the dimension of $C$ is $m=2$.

Wir have also the minimum distance $d(C) =3$.

The generator matrix of $C$ is \begin{equation*}G=\begin{pmatrix}1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \end{pmatrix}\end{equation*}

The canonical generator matrix is \begin{equation*}G'=\begin{pmatrix}1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 \end{pmatrix}\end{equation*}

And the canonical parity check matrix is
\begin{equation*}H'=\begin{pmatrix}1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1\end{pmatrix}\end{equation*}

Now there is the following question:

How many errors are at least detected at the code $C$ if $11100$ is received?

Could you give me a hint for that? 