# cyclic code definitions

## cyclic subspace

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

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:

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