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

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.