I consider primitive BCH codes, which are constructed as follows:
Choose positive integers $m$ and $r$, set $n=2^m-1$. Let $\alpha$ be a generator of the cyclic group $\mathbb{F}_{2^m}^*$. For each $1 \leq i < n$, let $m_i(x)$ be the minimal polynomial of $\alpha^i$ over $\mathbb{F}_2$. Then I consider the BCH code generated by the polynomial $lcm(m_1,\dotsc,m_r)$ (working modulo $x^n-1$).
It is known that the distance of such code is at least $r+1$. Can we bound the distance from above as well?
Conjecture: The distance is no more than $2(r+1)$ (maybe $2$ should be replaced with another small constant). Is it true?