Period-different m-sequences with at most four-valued cross correlation

We study the cross correlation between an m-sequence of period 2^m-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(2^l + 1) = 2ⁱ(mod 2ⁿ - 1) for some l and i, we prove that 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. 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 investigated here.