On the Nth 2-adic complexity of binary sequences identified with algebraic 2-adic integers

Abstract

We identify a binary sequence $S={\left(s_n\right)}_{n=0}^{\infty }$ with the 2-adic integer $G_S\left(2\right)=\sum _{n=0}^{\infty }{s}_n{2}^n$. In the case that $G_S\left(2\right)$ is algebraic over $Q$ of degree $d\ge 2$, we prove that the $N$th 2-adic complexity of $S$ is at least $\frac{N}{d}+O\left(1\right)$, where the implied constant depends only on the minimal polynomial of $G_S\left(2\right)$. This result is an analog of the bound of Mérai and the second author on the linear complexity of automatic sequences, that is, sequences with algebraic $G_S\left(X\right)$ over the rational function field $F_2\left(X\right)$.

We further discuss the most important case $d=2$ in both settings and explain that the intersection of the set of 2-adic algebraic sequences and the set of automatic sequences is the set of (eventually) periodic sequences. Finally, we provide some experimental results supporting the conjecture that 2-adic algebraic sequences can have also a desirable $N$th linear complexity and automatic sequences a desirable $N$th 2-adic complexity, respectively.