Example

Take g = (X + 1)(X³ + X + 1) Z/2Z to be a generator of a cyclic code of length 7. The corresponding check polynomial is

h = X³ + X² + 1.

Now, choose an information vector, say,

a = X.

It maps to code word

c = ag mod X⁷ - 1 = X⁵ + X⁴ + X³ + X.

Since

ch = X⁸ + X,

the polynomial of minimal degree in ch + (X⁷) is

X,

which coincides with a.