Linear Code/Examples/(2,2) of V(3,2)

Example of Linear Code

The set:

$C := \set {000, 001, 010, 011}$

is a linear $\tuple {2, 2}$-code of elements of $\Z_2$.

Proof

$(C \, 1): \quad$ Subtracting any of the codewords from any other gives a codeword beginning with $0$.

$(C \, 2): \quad$ As $0$ and $1$ are the only elements of $\Z_2$, then any multiple of any code is another element of $C$.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Example $6.3$