On the $N$th linear complexity of automatic sequences

TL;DR

This paper investigates the linear complexity of automatic sequences over finite fields, showing that some predictable sequences can have maximal linear complexity, with exact values computed for notable examples like Thue–Morse.

Contribution

It demonstrates that not all automatic sequences with high linear complexity are unpredictable and provides exact complexity values for key sequences.

Findings

Automatic sequences over finite fields can have linear complexity of order N.

Not all sequences with high linear complexity are unpredictable.

Exact linear complexities are computed for Thue–Morse and Rudin–Shapiro sequences.

Abstract

The $N$th linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large $N$th linear complexity. However, in this paper we show that for $q$-automatic sequences over $\mathbb{F}_q$ the converse is not true. We prove that any (not ultimately periodic) $q$-automatic sequence over $\mathbb{F}_q$ has $N$th linear complexity of order of magnitude $N$. For some famous sequences including the Thue--Morse and Rudin--Shapiro sequence we determine the exact values of their $N$th linear complexities. These are non-trivial examples of predictable sequences with $N$th linear complexity of largest possible order of magnitude.