Construction of sequences with high nonlinear complexity from a generalization of the Hermitian function field

TL;DR

This paper constructs sequences with high nonlinear complexity from a generalized Hermitian function field, leveraging properties of rational functions on maximal curves over finite fields.

Contribution

It introduces a method to generate sequences with high nonlinear complexity using a new class of function fields related to generalized Hermitian curves.

Findings

Sequences exhibit high nonlinear complexity.

Utilizes rational functions with small pole divisors.

Based on maximal curves over finite fields.

Abstract

For $r \geq 1$ an odd integer, we provide a sequence from the function field $\mathcal{F}_{q, r}$ of the maximal curve over $\mathbb{F}_{q^{2r}}$ defined by the affine equation $y^q+y=x^{q^r + 1}$. This sequence has high nonlinear complexity, and this fact comes from the existence of a rational function on $\mathcal{F}_{q, r}$ with pole divisor of small degree, and support in certain $q$ rational places.