Define a $B C H$ code of length $n$, where $n$ is odd, over the field of 2 elements with design distance $\delta$. Show that the minimum weight of such a code is at least $\delta$. [Results about the Vandermonde determinant may be quoted without proof, provided they are stated clearly.]

Let $\omega \in \mathbb{F}_{16}$ be a root of $X^{4}+X+1$. Let $C$ be the $\mathrm{BCH}$ code of length 15 with defining set $\left\{\omega, \omega^{2}, \omega^{3}, \omega^{4}\right\}$. Find the generator polynomial of $C$ and the rank of $C$. Determine the error positions of the following received words:

(i) $r(X)=1+X^{6}+X^{7}+X^{8}$,

(ii) $r(X)=1+X+X^{4}+X^{5}+X^{6}+X^{9}$.