Maximum-order complexity and $2$-adic complexity

TL;DR

This paper introduces a new method to analyze the $N$th $2$-adic complexity of finite-length binary sequences, establishing bounds related to maximum-order complexity and providing insights into sequence unpredictability.

Contribution

It presents the first theoretical approach linking $N$th maximum-order complexity with $N$th $2$-adic complexity for aperiodic sequences, including bounds and characterizations.

Findings

Lower bounds on $2$-adic complexity derived from maximum-order complexity.

Periodic sequences of maximal maximum-order complexity also have maximal $2$-adic complexity.

The bounds are sharp, demonstrated with $ ext{l}$-sequences.

Abstract

The $2$-adic complexity has been well-analyzed in the periodic case. However, we are not aware of any theoretical results on the $N$th $2$-adic complexity of any promising candidate for a pseudorandom sequence of finite length $N$ or results on a part of the period of length $N$ of a periodic sequence, respectively. Here we introduce the first method for this aperiodic case. More precisely, we study the relation between $N$th maximum-order complexity and $N$th $2$-adic complexity of binary sequences and prove a lower bound on the $N$th $2$-adic complexity in terms of the $N$th maximum-order complexity. Then any known lower bound on the $N$th maximum-order complexity implies a lower bound on the $N$th $2$-adic complexity of the same order of magnitude. In the periodic case, one can prove a slightly better result. The latter bound is sharp which is illustrated by the maximum-order complexity of $\ell$-sequences. The idea of the proof helps us to characterize the maximum-order complexity of periodic sequences in terms of the unique rational number defined by the sequence. We also show that a periodic sequence of maximal maximum-order complexity must be also of maximal $2$-adic complexity.