New pairs of m-sequences with 4-level cross-correlation

Let ω be a primitive element of GF(2n), where n≡0(mod4). Let d=(22k+2s+1-2k+1-1)/(2s-1), where n=2k, and s is such that 2s divides k. We prove that the binary m-sequences s(t)=tr(ωt) and s(dt) have a four-level cross-correlation function and give the distribution of the values.