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=(c₁,...,c_n) from C, the word (c_n,c₁,...,c_n-1) in Aⁿ 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=F₂ and n=3, the codewords contained in the (1,1,0)-cyclic code are precisely $(0,0,0),(1,1,0),(0,1,1)$ and $(1,0,1)$.

Trivial examples of cyclic codes are Aⁿ itself and the code containing only the zero codeword.

Cyclic Code at PlanetMath.