Standard Generator Matrix for Linear Code/Examples/(6, 3) code in Z2/Example
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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.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Example $6.13$