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

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 $p$.