A New Family of Binary Sequences via Elliptic Function Fields over Finite Fields of Odd Characteristics

TL;DR

This paper introduces a new family of binary sequences derived from cyclic elliptic function fields over finite fields of odd characteristic, extending previous constructions and providing bounds on their properties.

Contribution

It extends the construction of binary sequences using elliptic function fields to odd characteristic cases with explicit bounds on sequence parameters.

Findings

Sequences have length q+1+t and size q^{d-1}-1.

Balance and correlation bounds are explicitly derived.

Linear complexity is lower bounded by a formula involving q, t, and d.

Abstract

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 $\eta$ 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 $x\in\mathbb{R}$.