A Note on Hall’s Sextic Residue Sequence: Correlation Measure of Order $k$ and Related Measures of Pseudorandomness

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 Nth maximum order complexity of order of magnitude log p, which is the expected order of magnitude for a random sequence of length p.