Let $q$ be an odd prime power and let $\mathbb{F}_{q^2}$ be the finite field with $q^2$ elements. In this paper, we determine the differential spectrum of the power function $F(x)=x^{2q+1}$ over $\mathbb{F}_{q^2}$. When the characteristic of $\mathbb{F}_{q^2}$ is $3$, we also determine the value distribution of the Walsh spectrum of $F$, showing that it is $4$-valued, and use the obtained result to determine the weight distribution of a $4$-weight cyclic code.

翻译：令 $q$ 为奇素数幂，$\mathbb{F}_{q^2}$ 为包含 $q^2$ 个元素的有限域。本文确定了 $\mathbb{F}_{q^2}$ 上幂函数 $F(x)=x^{2q+1}$ 的差分谱。当 $\mathbb{F}_{q^2}$ 的特征为 $3$ 时，我们还确定了 $F$ 的沃尔什谱的值分布，证明其为 $4$ 值分布，并利用所得结果确定了一个 $4$ 重量循环码的重量分布。