Two Classes of Asymptotically Optimal Codebooks with Respect to the Welch Bound

Codebooks with small cross-correlation amplitude have extensive applications in many fields including code division multiple access (CDMA) communications systems, space-time codes, and compressed sensing. In this paper, we study an interesting character sum over finite fields and propose two constructions of complex codebooks from the operations of certain sets. The complex codebooks derived from these constructions are proved to be asymptotically optimal with respect to the Welch bound. The parameters of these codebooks are new and flexible.