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

Original · EN

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 2ᵐ-1 and the d-decimation of an m-sequence of shorter period 2ⁿ-1 for an even number m=2n. Assuming that d satisfies d(2ˡ+1)=2ⁱ(mod 2ⁿ-1) for some l and i, we prove the cross correlation takes exactly either three or four values, depending on 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.