  In this paper, we present explicit description on the additive characters,
multiplicative characters and Gauss sums over a local ring. As an application,
based on the additive characters and multiplicative characters satisfying
certain conditions, two new constructions of complex codebooks over a local
ring are introduced. With these two constructions, we obtain two families of
codebooks achieving asymptotically optimal with respect to the Welch bound.
It's worth mentioning that the codebooks constructed in this paper have new
parameters.
