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

Example of Syndrome Decoding on Linear $\tuple {6, 3}$-code in $\Z_2$
Let $C$ be the linear code:


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

Then the Syndrome Decoding of $100111$ yields $100110$.

Proof
The standard parity check table for $C$ is:


 * $P := \begin{pmatrix}

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

The syndrome decoding table $T$ for $C$ is:


 * $\begin{array} {cc}

000000 & 000 \\ 100000 & 110 \\ 010000 & 101 \\ 001000 & 011 \\ 000100 & 100 \\ 000010 & 010 \\ 000001 & 001 \\ 100001 & 111 \\ \end{array}$

By multiplying $P$ by $\paren {100111}^\intercal$, we get that the syndrome for $100111$ is $001$.

We find $000001$ in column $1$ and subtract it from $100111$.

We obtain $100110$.