Linear Code/Examples/Non-Linear Code/Subset of V(6,2)

Example of Non-Linear Code
Let $\map V {6, 2}$ denote the set of sequences of length $6$ of elements of $\Z_2$.

For all $x \in \Z_p$, let $\overline x$ denote $1 + x \pmod 2$ so that:
 * $\overline 0 = 1$
 * $\overline 1 = 0$

Let $C \subseteq \map V {6, 2}$ be the set of elements of $\map V {6, 2}$ of the form:
 * $x y z \overline x \overline y \overline z$

Then $C$ is not a linear code.

Proof
The elements of $C$ are:
 * $000111, 001110, 010101, 011100, 100011, 101010, 110001, 111000$

The minimum distance of $C$ is $2$.

We immediately note that:
 * $000111 + 001110 = 001001$

but that $001001 \notin C$.

So $C$ is not a linear code by definition.