  In this paper, we present two new constructions of complex codebooks with
multiplicative characters, additive characters and trace functions over finite
fields, and determine the maximal cross-correlation amplitude of these
codebooks. We prove that the codebooks we constructed are asymptotically
optimal with respect to the Welch bound. Moreover, in the first construction,
we generalize the result in [28]. In the second construction, we generalize the
results in [12], we can achieve Welch bound for any odd prime p, we also derive
the whole distribution of their inner products. The parameters of these
codebooks are new.
