Period-Different $m$-Sequences With At Most A Four-Valued Cross Correlation

TL;DR

This paper analyzes the cross correlation properties of period-different m-sequences, proving they take at most four values under certain conditions, and fully determining the distribution of these values.

Contribution

It provides a complete characterization of the cross correlation values for specific decimations of m-sequences, extending previous numerical observations.

Findings

Cross correlation takes exactly 3 or 4 values depending on gcd(l,n).

Distribution of correlation values is fully determined.

Confirms previous numerical phenomena and conjectures no other cases yield at most four values.

Abstract

In this paper, we follow the recent work of Helleseth, Kholosha, Johanssen and Ness to study the cross correlation between an $m$-sequence of period $2^m-1$ and the $d$-decimation of an $m$-sequence of shorter period $2^{n}-1$ for an even number $m=2n$. Assuming that $d$ satisfies $d(2^l+1)=2^i({\rm mod} 2^n-1)$ for some $l$ and $i$, we prove the cross correlation takes exactly either three or four values, depending on ${\rm gcd}(l,n)$ is equal to or larger than 1. The distribution of the correlation values is also completely determined. Our result confirms the numerical phenomenon Helleseth et al found. It is conjectured that there are no more other cases of $d$ that give at most a four-valued cross correlation apart from the ones proved here.