Generator polynomial of dual code?

Question

Let talk about Cyclic codes, if $C$ is an $[n, k]$ cyclic code generated by $g(x) $and and $h(x) = \frac {x^n−1}{g(x)}$. How can i proof that the dual code of $C$ is a cyclic $[n, n − k]$ code whose generator polynomial is $(x^k)h(x^{-1})$

Thank you in advance to any one who may be able to give me some ideas or answers.

Answer 1

It is clear that the dual code of $\mathcal{C}$ is generated by the polynomial $\sum_{j=0}^k h_{k-j}x^j$. And observe that

$$\sum_{j=0}^k h_{k-j}x^j=\sum_{j=0}^k h_jx^{k-j}=x^k\sum_{j=0}^k h_jx^{-j}=x^kh(x^{-1})$$