Compute the rank and minimum distance of the cyclic code with generator polynomial $g(X)=X^{3}+X+1$ and parity-check polynomial $h(X)=X^{4}+X^{2}+X+1$. Now let $\alpha$ be a root of $g(X)$ in the field with 8 elements. We receive the word $r(X)=X^{5}+X^{3}+X \quad\left(\bmod X^{7}-1\right)$. Verify that $r(\alpha)=\alpha^{4}$, and hence decode $r(X)$ using minimum-distance decoding.