Weight distributions of a class of cyclic codes from Fl-conjugates

Let $\Bbb F_{q^k}$ be a finite field and $\alpha$ a primitive element of $\Bbb F_{q^k}$, where $q=l^f$, $l$ is a prime power, and $f$ is a positive integer. Suppose that $N$ is a positive integer and $m_{g^{l^u}}(x)$ is the minimal polynomial of $g^{l^u}$ over $\Bbb F_q$ for $u=0, 1, \ldots, f-1$, where $g=\alpha^{-N}$. Let $\mathcal C$ be a cyclic code over $\Bbb F_q$ with check polynomial $$m_g(x)m_{g^l}(x) \cdots m_{g^{l^{f-1}}}(x).$$ In this paper, we shall present a method to determine the weight distribution of the cyclic code $\mathcal C$ in two cases: (1) $\gcd(\frac {q^k-1} {l-1}, N)=1$; (2) $l=2$ and $f=2$. Moreover, we will obtain a class of two-weight cyclic codes and a class of new three-weight cyclic codes.