A new family of binary sequences with a low correlation via elliptic curves
<|reference_start|>A new family of binary sequences with a low correlation via elliptic curves: 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.<|reference_end|>
@article{jin2024a,
    title={A new family of binary sequences with a low correlation via elliptic
  curves},
    author={Lingfei Jin, Liming Ma, Chaoping Xing and Runtian Zhu},
    journal={arXiv preprint arXiv:2407.18570},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.18570},
    primaryClass={math.NT cs.IT math.IT}
}