Constructions of Codebooks Asymptotically Achieving the Welch Bound With Additive Characters

Codebooks with small inner product correlation have wide applications in many fields, such as direct spread code-division multiple access communications, signal processing, and compressed sensing. It is extremely hard to produce optimal codebooks with respect to the Welch bound or the Levenstein bound. This letter focuses on constructing asymptotically optimal codebooks with additive characters of finite fields. Following the proposed constructions, two classes of asymptotically optimal codebooks with respect to the Welch bound are obtained. In addition, the resulting codebooks could have a small alphabet size.