= Solution
Let $\alpha$ be a primitive $n$th root of unity in an extension of $\mathbb F_q$, where $\gcd(n,q)=1$. A <BCH code> of length $n$, initial exponent $b$, and design distance $\delta$ is the cyclic code whose generator polynomial is the least common multiple over $\mathbb F_q$ of the minimal polynomials of
$$
\alpha^b,\alpha^{b+1},\ldots,\alpha^{b+\delta-2}.
$$
Thus every codeword polynomial $c(X)$ vanishes at these $\delta-1$ consecutive powers.
Suppose a nonzero codeword has weight $w<\delta$, with
$$
c(X)=\sum_{\ell=1}^w c_\ell X^{i_\ell},
$$
where the positions $i_\ell$ are distinct modulo $n$ and every $c_\ell$ is nonzero. The first $w$ root conditions give
$$
\sum_{\ell=1}^w c_\ell\alpha^{i_\ell(b+j)}=0,
\qquad j=0,1,\ldots,w-1.
$$
The coefficient matrix is a Vandermonde matrix in the distinct elements $\alpha^{i_1},\ldots,\alpha^{i_w}$, followed by multiplication of its columns by the nonzero factors $\alpha^{bi_\ell}$. Its determinant is therefore
$$
\left(\prod_{\ell=1}^w\alpha^{bi_\ell}\right)
\prod_{r<s}(\alpha^{i_s}-\alpha^{i_r})\ne0.
$$
Hence all $c_\ell$ would be zero, a contradiction. The minimum distance consequently satisfies $d\geq\delta$.
A code of minimum distance $d$ detects every pattern of at most $d-1$ errors and uniquely corrects every pattern of at most $\lfloor(d-1)/2\rfloor$ errors, because Hamming balls of that radius are disjoint. A BCH code of design distance $\delta$ is therefore guaranteed to detect $\delta-1$ errors and to correct
$$
\left\lfloor\frac{\delta-1}{2}\right\rfloor
$$
errors; its actual capabilities may be larger if $d>\delta$.
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