The weight distribution of a family of p-ary cyclic codes

Let m, k be positive integers, p be an odd prime and $\pi$ be a primitive element of ${\mathbb{F}}_{p^m}$. In this paper, we determine the weight distribution of a family of cyclic codes ${\mathcal{C}}_t$ over ${\mathbb{F}}_p$, whose duals have two zeros ${\pi }^{-t}$ and $-{\pi }^{-t}$, where $t$ satisfies $t\equiv \frac{p^k+1}{2}{p}^{\tau }(\mathrm{m}\mathrm{o}\mathrm{d}\frac{p^m-1}{2})$ for some $\tau \in {0,1,\cdots, m-1}$.