Standard Generator Matrix for Linear Code/Examples/(6, 3) code in Z2/Example

Example of Error Detection in Linear $\tuple {6, 3}$ code in $\Z_2$
Let $C$ be the linear code:

$C = \set {000000, 100110, 010101, 110011, 001011, 101101, 011110, 111000}$

The received word $100111$ has $1$ transmission error and so can be corrected.

The received word $100001$ has $2$ transmission errors and so cannot be corrected.

Proof
$100111$ is at a distance $1$ from the codeword $100110$.

As the minimum distance of $C$ is $3$, there is no codeword that is closer than that.

Hence the transmitted codeword can be inferred as being $100110$.

$100001$ is at a distance $2$ from the codewords $000000$, $110011$ and $101101$.

Hence, while it is detected as having $2$ transmission errors, it is not possible to uniquely determine what the transmitted codeword could have been.