  Binary $m$-sequences are ones with the largest period $n=2^m-1$ among the
binary sequences produced by linear shift registers with length $m$. They have
a wide range of applications in communication since they have several desirable
pseudorandomness such as balance, uniform pattern distribution and ideal
(classical) autocorrelation. In his reseach on arithmetic codes, Mandelbaum
\cite{9Mand} introduces a 2-adic version of classical autocorrelation of binary
sequences, called arithmetic autocorrelation. Later, Goresky and Klapper
\cite{3G1,4G2,5G3,6G4} generalize this notion to nonbinary case and develop
several properties of arithmetic autocorrelation related to linear shift
registers with carry. Recently, Z. Chen et al. \cite{1C1} show an upper bound
on arithmetic autocorrelation of binary $m$-sequences and raise a conjecture on
absolute value distribution on arithmetic autocorrelation of binary
$m$-sequences.
