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 $$(0,0,0), (1,1,0), (0,1,1)\text{ and }(1,0,1).$$

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