  Professor Cunsheng Ding gave cyclotomic constructions of cyclic codes with
length being the product of two primes. In this paper, we study the cyclic
codes of length $n=2^e$ and dimension $k=2^{e-1}$. Clearly, Ding's construction
is not hold in this place. We describe two new types of generalized cyclotomy
of order two, which are different from Ding's. Furthermore, we study two
classes of cyclic codes of length $n$ and dimension $k$. We get the enumeration
of these cyclic codes. What's more, all of the codes from our construction are
among the best cyclic codes. Furthermore, we study the hull of cyclic codes of
length $n$ over $\mathbb{F}_q$. We obtain the range of $\ell=\dim({\rm
Hull}(C))$. We construct and enumerate cyclic codes of length $n$ having hull
of given dimension.
