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

Abstract: Let m, k be positive integers, p be an odd prime and $\pi$ be a primitive element of $\mathbb{F}_{p<sup>m}$. 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 $\pi<sup>{-t}$ and $-\pi<sup>{-t}$, where $t$ satisfies $$t\equiv \frac{p^{k+1}{2}p^\tau} \ ({\rm mod}\ \frac{p^m-1}{2})$$ for some $\tau \in {0,1,\cdots, m-1}$.