Syndrome/Examples/(6, 3) code in Z2

Example of Syndrome
Let $C$ be the linear $\tuple {6, 3}$-code in $\Z_2$ whose standard generator matrix $G$ is given by:


 * $G := \begin{pmatrix}

1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \end{pmatrix}$

The syndrome of $100000$ is $110$

The syndrome of $110011$ is $000$.

Proof
From Standard Parity Check Matrix on the given linear $\tuple {6, 3}$-code in $\Z_2$:


 * $P := \begin{pmatrix}

1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{pmatrix}$

where $P$ is the standard parity check matrix of $C$.

Thus: