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 $k\mathrm{<mo>\; |\; </mo>}\left(q-1\right)$

${\mathbb{F}}_p$

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 ${\mathbb{F}}_q^{\ast }$

${\text{Tr}}_{p^m|q}(\cdot )$

trace function maps ${\mathbb{F}}_{p^m}$ element to ${\mathbb{F}}_q$ element

${ϵ}_k$

primitive k-th root of unity

$f_k(\cdot )$

mapping function maps ${\mathbb{F}}_q$ element to ${\mathbb{F}}_k$ element

$\mathcal{S}$

proposed sequence in this paper

n

period of a sequence

${\text{R}}_{\mathcal{S}}(\cdot )$

autocorrelation of a sequence $\mathcal{S}$

${\text{R}}_{\widehat{\mathcal{S}}\mathrm{,}\mathcal{S}}(\cdot )$

cross-correlation between the sequences $\mathcal{S}$ and $\widehat{\mathcal{S}}$