On the $N$th $2$-adic complexity of binary sequences identified with algebraic $2$-adic integers

Abstract

We identify a binary sequence $\mathcal{S}=(s_n)_{n=0}^\infty$ with the $2$-adic integer $G_\mathcal{S}(2)=\sum\limits_{n=0}^\infty s_n2^n$. In the case that $G_\mathcal{S}(2)$ is algebraic over $\mathbb{Q}$ of degree $d\ge 2$, we prove that the $N$th $2$-adic complexity of $\mathcal{S}$ is at least $\frac{N}{d}+O(1)$, where the implied constant depends only on the minimal polynomial of $G_\mathcal{S}(2)$. This result is an analog of the bound of M\'erai and the second author on the linear complexity of automatic sequences, that is, sequences with algebraic $G_\mathcal{S}(X)$ over the rational function field $\mathbb{F}_2(X)$. 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.