I have to prove that, given a narrow-sense binary bch code, its minimum distance (i.e. Hamming distance) is always odd.
Any idea/hint on how to start the proof?
Edit: I write down the definitions we use:
Definition ((primitive) BCH codes) Let $\mathbb{F}$ be the splitting field of $x^n-1$ on $\mathbb{F}_q$ and let $\alpha \in \mathbb{F}$ be a primitive $n$-root of unity. Let $S=\{m, m+1, \ldots, m+\delta -2\}$ be such that $$0\le m \le \ldots \le m+\delta -2\le n-1.$$ If $C$ is the $[n,k,d]_q$ cyclic code with defining set $S$, we say that $C$ is a (primitive) BCH code of defining distance $\delta$.
Definition (binary narrow-sense BCH code) For any positive integer $a$ and $d_0\le 2^a-2$ we can consider a (primitive) BCH code $C$ with parameters $$q=2 \qquad n=2^a-1 \qquad m=1 \qquad \delta =d_0.$$ We say that $C$ is the binary narrow-sense (primitive) BCH code of length $2^a-1$ and designed distance $\delta$.