Probabilistic results on the 2-adic complexity

Abstract

This work is devoted to solving some closely related open problems on the average and asymptotic behavior of the 2-adic complexity of binary sequences. First, for fixed N, we prove that the expected value \(E^{\text {2-adic}}_N\) of the 2-adic complexity over all binary sequences of length N is close to \(\frac{N}{2}\) and the deviation from \(\frac{N}{2}\) is at most of order of magnitude \(\log (N)\). More precisely, we show that

$$\begin{aligned} \frac{N}{2}-1 \le E^{\text {2-adic}}_N= \frac{N}{2}+O(\log (N)). \end{aligned}$$

We also prove bounds on the expected value of the Nth rational complexity. Our second contribution is to prove for a random binary sequence \(\mathcal {S}\) that the Nth 2-adic complexity satisfies with probability 1

$$\begin{aligned} \lambda _{\mathcal {S}}(N)=\frac{N}{2}+O(\log (N)) \, \hbox { for all}\ N. \end{aligned}$$