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

摘要

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 .