I start to learn about Reed-Solomon code and I find this question but have hard time to solve it -$$$$ Let $\alpha \in F_{2^m}$ be a primitive element. Let $C_{BCH}$ be a code over $F_2$ of length $n = 2^m- 1$ with a parity-check matrix, $H_{BCH} = $
\begin{pmatrix} 1 & \alpha & \alpha^2 & ... & \alpha^{n-1}\\ 1 & \alpha^3 & \alpha^6 &... & \alpha^{n-1} \end{pmatrix}
Let $C_{RS}$ be a conventional Reed-Solomon code over $F_{2^m}$ with parameters [N, K, D], $N = n = 2^m-1$, such that $$C_{BCH} = C_{RS} \cap F^N_2 $$
Write a full-rank parity-check matrix for $C_{RS}$ if we know that D = 5 and $\alpha^3$ is a root of $C_{RS}$.