On cyclic codes of length $2^e$ over finite fields

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.