Random properties of the highest level sequences of primitive sequences over [[Z.sub.[2.sup.e]]

Using the estimates of the exponential sums over Galois rings, we discuss the random properties of the highest level sequences [a.sub.e-1] of primitive sequences generated by a primitive polynomial of degree n over [Z.sub.[2.sup.e]]. First we obtain an estimate of 0, 1 distribution in one period of [a.sub.e-1]. On the other hand, we give an estimate of the absolute value of the autocorrelation function |[C.sub.N] (h)| of [a.sub.e-1], which is less than [2.sup.e-1] ([2.sup.e-1] - 1) [square root of (3 [2.sup.2e] - 1)) [2.sup.n/2] + 2.sup.e-1] for h [not equal to] 0. Both results show that the larger n is, the more random [a.sub.e-1] will be. Index Terms--0, 1 distribution, autocorrelation function, exponential sum over Galois ring, highest level sequence, primitive sequence.