通过奇特征有限域上椭圆函数域构造的新型二进制序列族 (A New Family of Binary Sequences via Elliptic Function Fields over Finite Fields of Odd Characteristics)

Motivated by the constructions of binary sequences by utilizing the cyclic elliptic function fields over the finite field $\mathbb{F}_{2^{n}}$ by Jin \textit{et al.} in [IEEE Trans. Inf. Theory 71(8), 2025], we extend the construction to the cyclic elliptic function fields with odd characteristic by using the quadratic residue map $η$ instead of the trace map used therein. For any cyclic elliptic function field with $q+1+t$ rational points and any positive integer $d$ with $\gcd(d, q+1+t)=1$, we construct a new family of binary sequences of length $q+1+t$, size $q^{d-1}-1$, balance upper bounded by $(d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+d,$ the correlation upper bounded by $(2d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+2d$ and the linear complexity lower bounded by $\frac{q+1+2t-d-(d+1)\cdot\lfloor2\sqrt{q}\rfloor}{d+d\cdot\lfloor2\sqrt{q}\rfloor}$ where $\lfloor x\rfloor$ stands for the integer part of $x\in\mathbb{R}$.

翻译：受 Jin 等人在 [IEEE Trans. Inf. Theory 71(8), 2025] 中利用有限域 $\mathbb{F}_{2^{n}}$ 上的循环椭圆函数域构造二进制序列的启发，我们将该构造推广至奇特征循环椭圆函数域，并使用二次剩余映射 $η$ 替代原文中的迹映射。对于任意具有 $q+1+t$ 个有理点的循环椭圆函数域，以及满足 $\gcd(d, q+1+t)=1$ 的任意正整数 $d$，我们构造了一族长度为 $q+1+t$、规模为 $q^{d-1}-1$ 的新型二进制序列，其平衡度上界为 $(d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+d$，相关值上界为 $(2d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+2d$，线性复杂度下界为 $\frac{q+1+2t-d-(d+1)\cdot\lfloor2\sqrt{q}\rfloor}{d+d\cdot\lfloor2\sqrt{q}\rfloor}$，其中 $\lfloor x\rfloor$ 表示 $x\in\mathbb{R}$ 的整数部分。