  Under research for near sixty years,
Bose-$\!$Ray-$\!$Chaudhuri-$\!$Hocquenghem(BCH) codes have played increasingly
important roles in many applications such as communication systems, data
storage and information security. However, the dimension and minimum distance
of BCH codes are seldom solved until now because of their intractable
characteristics. The objective of this paper is to study the dimensions of some
BCH codes of length $n=2^m+1$ with $m=2t+1$, $4t+2$, $8t+4$ and $m\geq 10$.
Some new techniques are employed to investigate coset leaders modulo $n$. For
each type of $m$ above, the first five largest coset leaders modulo $n$ are
determined, the dimension of some BCH codes of length $n$ with designed
distance $\delta>2^{\lceil \frac{m}{2} \rceil}$ is presented. These new
techniques and results may be helpful to study other families of cyclic codes
over finite fields.
