In the polynomial ring Z/2Z[X] we consider the polynomial Xn - 1 with n > 1. Let S denote the residue class ring Z/2Z[X]/(Xn - 1). This ring has the structure of vector space over the field Z/2Z with basis 1 , ..., Xn-1. So each element of S can be represented by the vector of coefficients with respect to this basis, and vice versa:

a = a0 + a1X + ··· + an-1Xn-1    <->    a = (a0, a1, ..., an-1).

The polynomial Xn - 1 is reducible for n > 1: it is divisible by X - 1.

Definition

Let g be a divisor of Xn - 1 in Z/2Z[X]. The image under the linear map

Z/2Z[X] -> S,     a -> ag + (Xn - 1)

is called the cyclic code of length n generated by g.

Let l be the degree of g and write k = n-l. The elements g, ..., Xk-1g form a basis for the image space C of the map. So the dimension of C is equal to n-l = k. The space C is called the code generated by g. The polynomial g is known as the generator of C.

The quotient (Xn-1)/g is called the check polynomial of C. On the next page, we discuss its use.