Random properties of the highest level sequences of primitive sequences over Z(2e)

Using the estimates of the exponential sums over Galois rings, we discuss the random properties of the highest level sequences α_e-1 of primitive sequences generated by a primitive polynomial of degree n over Z(2^e). First we obtain an estimate of 0, 1 distribution in one period of α_e-1. On the other hand, we give an estimate of the absolute value of the autocorrelation function |C_N(h)| of α_e-1, which is less than 2^e-1(2^e-1-1)√3(2^2e-1)2ⁿ2/+2^e-1 for h≠0. Both results show that the larger n is, the more random α_e-1 will be.