On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients

Abstract

Let \begin{document}$ p $\end{document} be a prime and let \begin{document}$ n $\end{document} be an integer with \begin{document}$ (n, p) = 1 $\end{document} . The Fermat quotient \begin{document}$ q_p(n) $\end{document} is defined as \begin{document}$ q_p(n)\equiv \frac{n^{p-1}-1}{p} \ (\bmod\ p), \quad 0\leq q_p(n)\leq p-1. $\end{document} We also define \begin{document}$ q_p(kp) = 0 $\end{document} for \begin{document}$ k\in \mathbb{Z} $\end{document} . Chen, Ostafe and Winterhof constructed the binary sequence \begin{document}$ E_{p^2} = \left(e_0, e_1, \cdots, e_{p^2-1}\right)\in \{0, 1\}^{p^2} $\end{document} as \begin{document}$ \begin{equation*} \begin{split} e_{n} = \left\{\begin{array}{ll} 0, & \hbox{if }\ 0\leq \frac{q_p(n)}{p} and studied the well-distribution measure and correlation measure of order \begin{document}$ 2 $\end{document} by using estimates for exponential sums of Fermat quotients. In this paper we further study the correlation measures of the sequence. Our results show that the correlation measure of order \begin{document}$ 3 $\end{document} is quite good, but the \begin{document}$ 4 $\end{document} -order correlation measure of the sequence is very large.