Linear Code/Examples

Examples of Linear Codes

Elements of $\map V {3, 2}$

The master code $\map V {3, 2}$ of length $3$ of elements of $\Z_2$ consists of:

$\map V {3, 2} = \set {000, 001, 010, 011, 100, 101, 110, 111}$

Elements of $\map V {2, 3}$

The master code $\map V {2, 3}$ of length $2$ of elements of $\Z_3$ consists of:

$\map V {2, 3} = \set {00, 01, 02, 10, 11, 12, 20, 21, 22}$

Linear $\tuple {2, 2}$-code of $\map V {3, 2}$

The set:

$C := \set {000, 001, 010, 011}$

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

Linear $\tuple {1, 1}$-code of $\map V {2, 3}$

The set:

$C := \set {11, 22, 33}$

is a linear $\tuple {2, 1}$-code of elements of $\Z_3$.

Examples of Non-Linear Codes

Subset of $\map V {6, 2}$ which is not a 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.