In the realm of modern digital communication, cryptography, and signal processing, binary sequences with a low correlation properties play a pivotal role. In the literature, considerable efforts have been dedicated to constructing good binary sequences of various lengths. As a consequence, numerous constructions of good binary sequences have been put forward. However, the majority of known constructions leverage the multiplicative cyclic group structure of finite fields $\mathbb{F}_{p^n}$, where $p$ is a prime and $n$ is a positive integer. Recently, the authors made use of the cyclic group structure of all rational places of the rational function field over the finite field $\mathbb{F}_{p^n}$, and firstly constructed good binary sequences of length $p^n+1$ via cyclotomic function fields over $\mathbb{F}_{p^n}$ for any prime $p$ \cite{HJMX24,JMX22}. This approach has paved a new way for constructing good binary sequences. Motivated by the above constructions, we exploit the cyclic group structure on rational points of elliptic curves to design a family of binary sequences of length $2^n+1+t$ with a low correlation for many given integers $|t|\le 2^{(n+2)/2}$. Specifically, for any positive integer $d$ with $\gcd(d,2^n+1+t)=1$, we introduce a novel family of binary sequences of length $2^n+1+t$, size $q^{d-1}-1$, correlation bounded by $(2d+1) \cdot 2^{(n+2)/2}+ |t|$, and a large linear complexity via elliptic curves.

翻译：在现代数字通信、密码学和信号处理领域，具有低相关性质的二进制序列发挥着关键作用。已有文献中，研究者们投入了大量精力来构造各种长度的优良二进制序列，因此提出了众多优良二进制序列的构造方法。然而，大多数已知构造都利用了有限域 $\mathbb{F}_{p^n}$ 的乘法循环群结构，其中 $p$ 为素数，$n$ 为正整数。最近，作者们利用有限域 $\mathbb{F}_{p^n}$ 上有理函数域所有有理位置的循环群结构，首次通过 $\mathbb{F}_{p^n}$ 上的分圆函数域为任意素数 $p$ 构造出了长度为 $p^n+1$ 的优良二进制序列 \cite{HJMX24,JMX22}。该方法为构造优良二进制序列开辟了新途径。受上述构造的启发，我们利用椭圆曲线有理点上的循环群结构，设计了一族长度为 $2^n+1+t$ 且具有低相关性的二进制序列，其中整数 $|t|\le 2^{(n+2)/2}$ 可在较大范围内给定。具体而言，对于任意满足 $\gcd(d,2^n+1+t)=1$ 的正整数 $d$，我们通过椭圆曲线引入了一类新型二进制序列族，其长度为 $2^n+1+t$，规模为 $q^{d-1}-1$，相关性以 $(2d+1) \cdot 2^{(n+2)/2}+ |t|$ 为界，并具有较高的线性复杂度。