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 Fpⁿ, 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 Fpⁿ, and firstly constructed good binary sequences of length pⁿ+1 via cyclotomic function fields over Fpⁿ for any prime p {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 2ⁿ+1+t with a low correlation for many given integers |t|≤ 2(n+2)/2. Specifically, for any positive integer d with (d,2ⁿ+1+t)=1, we introduce a novel family of binary sequences of length 2ⁿ+1+t, size qᵈ⁻¹-1, correlation bounded by (2d+1) · 2(n+2)/2+ |t|, and a large linear complexity via elliptic curves.