A class of binary sequences with two-valued cross correlations

Let $p$ be an odd prime with $p\equiv 7(\mathrm{mod}12)$, $\frac{p-1}{2}$ be the least integer such that $2^{\frac{p-1}{2}}\equiv 1(\mathrm{mod}p)$, and $q=2^{\frac{p-1}{2}}$. Let $\alpha$ be a primitive element of the finite field ${\mathbb{F}}_q$ and $\beta ={\alpha }^{\frac{q-1}{p}}$. Suppose that $\sigma =\sum _{i=0}^2{\beta }^{m{\zeta }_3^i}\in {\mathbb{F}}_q^{\ast }$, where $m\in {\mathbb{F}}_p^{\ast }$ and ${\zeta }_3$ is a $3$rd root of unity in ${\mathbb{F}}_p$. Let $\{u_i\}=({\mathrm{Tr}}_{q/2}(\sigma {\beta }^i){)}_{i=0}^{q-2}$ be a binary sequence of period $q-1$. In this paper, we obtained the cross correlation distribution between two sequences $\{u_i\}$ and its $\frac{q-1}{p}$-decimation sequence, which is two-valued.