cyclic code

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 $c=(c_1,\mathrm{\dots },c_n)$ from $C$, the word $(c_n,c_1,\mathrm{\dots },c_{n-1})\in A^n$ obtained by a right shift of 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 shifts of its .

For example, if $A={𝔽}_2$ 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.