# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021008

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

 Research Center for Number Theory and Its Applications, School of Mathematics, Northwest University, Xi'an 710127, China

* Corresponding author: Huaning Liu

Received  October 2020 Revised  February 2021 Published  April 2021

Let
 $p$
be a prime and let
 $n$
be an integer with
 $(n, p) = 1$
. The Fermat quotient
 $q_p(n)$
is defined as
 $q_p(n)\equiv \frac{n^{p-1}-1}{p} \ (\bmod\ p), \quad 0\leq q_p(n)\leq p-1.$
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.
Citation: Huaning Liu, Xi Liu. On the correlation measures of orders $3$ and $4$ of binary sequence of period $p^2$ derived from Fermat quotients. Advances in Mathematics of Communications, doi: 10.3934/amc.2021008
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