Define a BCH 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 van der Monde determinant may be quoted without proof, provided they are stated clearly.]

Consider a BCH code of length 31 over the field of 2 elements with design distance 8 . Show that the minimum distance is at least 11. [Hint: Let $\alpha$ be a primitive element in the field of $2^{5}$ elements, and consider the minimal polynomial for certain powers of $\left.\alpha .\right]$