Pseudorandom sequences derived from automatic sequences

TL;DR

This paper surveys the pseudorandom properties of automatic sequences like Thue-Morse and Rudin-Shapiro, analyzing their strengths and weaknesses for applications, and explores subsequences that retain good features while avoiding bad ones.

Contribution

It provides a comprehensive analysis of the pseudorandomness and non-randomness of automatic sequences and their subsequences across various measures.

Findings

Automatic sequences have high linear complexity and low well-distribution measure.

Certain subsequences maintain good pseudorandom properties while avoiding undesirable patterns.

The paper discusses behaviors under measures like correlation, expansion complexity, and normality.

Abstract

Many automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence, have some desirable features of pseudorandomness such as a large linear complexity and a small well-distribution measure. However, they also have some disastrous properties in view of certain applications. For example, the majority of possible binary patterns never appears in automatic sequences and their correlation measure of order 2 is extremely large. Certain subsequences, such as automatic sequences along squares, may keep the good properties of the original sequence but avoid the bad ones. In this survey we investigate properties of pseudorandomness and non-randomness of automatic sequences and their subsequences and present results on their behaviour under several measures of pseudorandomness including linear complexity, correlation measure of order $k$, expansion complexity and normality. We also mention some analogs for finite fields.