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

TL;DR

The paper studies the $N$th $2$-adic complexity of binary sequences identified with $2$-adic integers when $G_\mathcal{S}(2)$ is algebraic of degree $d\ge 2$, proving lower bounds $\lambda_{\mathcal{S}}(N) \ge \frac{N}{d}+O(1)$ and providing matching upper bounds up to constants dependent on the defining polynomial. The authors adapt ideas from automatic sequences and employ $2$-adic and Hensel-type lifting arguments to derive the bounds, with a detailed analysis in the quadratic case ($d=2$) that connects $2$-adic algebraic sequences to automatic sequences and shows their intersection consists of (eventually) periodic sequences. They also discuss the implications for random-like behavior and provide experimental evidence suggesting that suitable $2$-adic algebraic sequences can exhibit favorable $N$th linear complexity, while automatic sequences can exhibit favorable $N$th $2$-adic complexity. The work highlights structural differences and potential crossovers between $2$-adic and automatic frameworks and raises questions about the coexistence of high linear and $2$-adic complexities in respective classes.

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 Mérai 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.

Figures (2)

• Figure 1: The $N$th linear complexity of $\mathcal{S}$ with $G_{\mathcal{S}}(2)=\sqrt{17}$ for $N\le 100$
• Figure 2: The $N$th linear complexity of $\mathcal{S}$ with $G_{\mathcal{S}}(2)=\sqrt{-7}$ for $N\le 100$

Theorems & Definitions (3)

• Theorem 1
• Lemma 1
• Lemma 2