= Solution
A binary <cyclic code> of odd length $n$ is an ideal of $\mathbb F_2[X]/(X^n-1)$. For a primitive $n$th root $\alpha$, its defining set is the set of powers $\alpha^i$ at which every code polynomial vanishes. A <BCH code> of design distance $\delta$ has $\delta-1$ consecutive powers
$$
\alpha^b,\alpha^{b+1},\ldots,\alpha^{b+\delta-2}
$$
in its defining set.
If a nonzero codeword had weight $w<\delta$, write it as $c(X)=\sum_{j=1}^wc_jX^{i_j}$. Evaluation at $w$ consecutive defining roots gives a homogeneous Vandermonde system in the nonzero values $c_j\alpha^{bi_j}$. Its determinant is nonzero because the support elements $\alpha^{i_j}$ are distinct. Thus every $c_j$ would vanish, a contradiction. This proves the <BCH bound>
$$
\boxed{d_{\min}\geq\delta}.
$$
Back to article page
