A note on Hall's sextic residue sequence: correlation measure of order $k$ and related measures of pseudorandomness

TL;DR

This paper analyzes Hall's sextic residue sequence, demonstrating that its correlation measure of order k closely resembles that of a random sequence, and establishes bounds on its maximum order complexity, highlighting its pseudorandom properties.

Contribution

It provides a detailed analysis of the correlation measure of order k for Hall's sextic residue sequence and derives bounds on its maximum order complexity, enhancing understanding of its pseudorandomness.

Findings

Correlation measure of order k is approximately p^{1/2}

Lower bound on maximum order complexity is about log p

Sequence exhibits pseudorandom features similar to random sequences

Abstract

It is known that Hall's sextic residue sequence has some desirable features of pseudorandomness: an ideal two-level autocorrelation and linear complexity of the order of magnitude of its period $p$. Here we study its correlation measure of order $k$ and show that it is, up to a constant depending on $k$ and some logarithmic factor, of order of magnitude $p^{1/2}$, which is close to the expected value for a random sequence of length $p$. Moreover, we derive from this bound a lower bound on the $N$th maximum order complexity of order of magnitude $\log p$, which is the expected order of magnitude for a random sequence of length $p$.