The cross correlation distribution of a p-ary m -sequence of period pm−1pm−1 and its decimated sequences by (pk+1)(pm+1)/4(pk+1)(pm+1)/4

Let p be an odd prime, m and k   be positive integers such that s=m/es=m/e is odd and k/ek/e is even, where e=gcd⁡(m,k)e=gcd⁡(m,k). In this paper, the value distribution of the exponential sum ∑x∈Fpmχ(αx(pk+1)(pm+1)/4+βx)∑x∈Fpmχ(αx(pk+1)(pm+1)/4+βx) is completely determined with pm≡1(mod4). As applications, when pm≡1(mod4), the distribution of the cross correlation between a p-ary m  -sequence (st)(st) of period pm−1pm−1 and its decimated sequence (sdt)(sdt) with d=(pk+1)(pm+1)/4d=(pk+1)(pm+1)/4 is determined. The sequence family FF is constructed and the cross correlation distribution among sequences in the family FF is also determined. Our result shows that the maximal magnitude of the correlation value is low when e=1e=1.