Let $C$ be a linear code over a finite field $A$ of block length $n$. $C$ is called a cyclic code, if for every codeword $c=(c_{1},\ldots,c_{n})$ from $C$, the word $(c_{n},c_{1},\ldots,c_{{n-1}})\in A^{n}$ obtained by a cyclic right shift of components is also a codeword from $C$.

Sometimes, $C$ is called the $c$-cyclic code, if $C$ is the smallest cyclic code containing $c$, or, in other words, $C$ is the linear code generated by $c$ and all codewords obtained by cyclic shifts of its components.

For example, if $A=\mathbb{F}_{2}$ and $n=3$, the codewords contained in the $(1,1,0)$-cyclic code are precisely

Trivial examples of cyclic codes are $A^{n}$ itself and the code containing only the zero codeword.