Arithmetic autocorrelation distribution of binary $m$-sequences

TL;DR

This paper investigates the distribution of arithmetic autocorrelation in binary m-sequences, which are crucial in communication systems for their pseudorandom properties, and explores bounds and conjectures related to their autocorrelation behavior.

Contribution

It provides an analysis of the arithmetic autocorrelation distribution of binary m-sequences, extending previous work and addressing conjectures on their autocorrelation properties.

Findings

Established bounds on arithmetic autocorrelation values.

Analyzed the distribution pattern of autocorrelation in binary m-sequences.

Proposed conjectures on the absolute value distribution of autocorrelation.

Abstract

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.