cyclic code definitions

cyclic subspace

A subspace $S$ of $V_n(F)$ is a cyclic subspace if whenever

$$
(a_1a_2a_3a_4…a_n) \in S \text{ then } (a_na_1a_2a_3…a_n-1) \in S
$$

In other words, $S$ is a subspace and for each vector $a \in S$, every cyclic shift of a is also in $S$.

cyclic code

A linear code $C$ is a cyclic code if $C$ is a cyclic subspace.
Examples:

1. $S={(0000)} \subseteq V_4(Z_2)$
2. $S={(0000),(1111)} \subseteq V_4(Z_2)$
3. $S={(0000),(1111),(1100),(0110),(0011),(1001),(1010),(0101)} \subseteq V_4(Z_2)$