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?
The answer appears exactly in Introduction to Coding Theory by van Lint.
Theorem 6.6.13 there proves that if the designed distance is of the form $2^l-1$ for some $l$ then the distance equals the designed distance exactly.
Corollary 6.6.14 then takes an arbitrary primitive BCH code, finds the largest primitive BCH code of the form of Theorem 6.6.13 and uses the codeword with lowest hamming weight in this subcode to show that for a general BCH code we have $d \leq 2r-1$ ($d$ is the distance, $r$ is the designed distance).
Thus, in general, $r \leq d \leq 2r-1$.
(van Lint takes $g(x) := lcm(m_1(x),\dotsc,m_{r-1}(x))$ as a generator).