List of symbols p odd prime number m positive integer, mainly denotes the extension degree ${m}^{\prime }$ one of the factors of m q power of the odd prime $q={p}^{{m}^{\prime }}$ k prime factor of $q-1$ such as $k|\left(q-1\right)$ ${\mathbb{F}}_{p}$ odd characteristic prime field ${\mathbb{F}}_{{p}^{m}}$ an extension field (base p and extension degree m) ${\mathbb{F}}_{q}$ sub extension field ${\mathbb{F}}_{q}^{\ast }$ multiplicative group of ${\mathbb{F}}_{q}$ , excluding the 0 such as ${\mathbb{F}}_{q}^{\ast }={\mathbb{F}}_{q}-\left\{0\right\}$ $f\left(x\right)$ a primitive polynomial $\omega$ root of an irreducible polynomial g generator of a group A an arbitrary element in ${\mathbb{F}}_{q}^{\ast }$ ${\text{Tr}}_{{p}^{m}|q}\left(\cdot \right)$ trace function maps ${\mathbb{F}}_{{p}^{m}}$ element to ${\mathbb{F}}_{q}$ element ${ϵ}_{k}$ primitive k-th root of unity ${f}_{k}\left(\cdot \right)$ mapping function maps ${\mathbb{F}}_{q}$ element to ${\mathbb{F}}_{k}$ element $\mathcal{S}$ proposed sequence in this paper n period of a sequence ${\text{R}}_{\mathcal{S}}\left(\cdot \right)$ autocorrelation of a sequence $\mathcal{S}$ ${\text{R}}_{\stackrel{^}{\mathcal{S}},\mathcal{S}}\left(\cdot \right)$ cross-correlation between the sequences $\mathcal{S}$ and $\stackrel{^}{\mathcal{S}}$