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

Abstract

Let $ p $ be a prime and let $ n $ be an integer with $ (n, p) = 1 $. The Fermat quotient $ q_p(n) $ 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 $ q_p(kp) = 0 $ for $ k\in \mathbb{Z} $. Chen, Ostafe and Winterhof constructed the binary sequence $ E_{p^2} = \left(e_0, e_1, \cdots, e_{p^2-1}\right)\in \{0, 1\}^{p^2} $ as$ \begin{equation*} \begin{split} e_{n} = \left\{\begin{array}{ll} 0, & \hbox{if }\ 0\leq \frac{q_p(n)}{p}<\frac{1}{2}, \\ 1, & \hbox{if }\ \frac{1}{2}\leq \frac{q_p(n)}{p}<1, \end{array} \right. \end{split} \end{equation*} $and studied the well-distribution measure and correlation measure of order $ 2 $ 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 $ 3 $ is quite good, but the $ 4 $-order correlation measure of the sequence is very large.