Incomplete exponential sums over galois rings with applications to some binary sequences derived from Z2l

An upper bound for the incomplete exponential sums over Galois rings is derived explicitly. Based on the incomplete exponential sums, we analyze the partial period properties of some binary sequences derived from Z₂^l in detail, such as the Kerdock-code binary sequences and the highest level sequences of primitive sequences over Z₂^l. The results show that the partial period distributions and the partial period independent r-pattern distributions of these binary sequences are asymptotically uniform. Nontrivial upper bounds for the aperiodic autocorrelation of these sequences are also given.