Two classes of nearly optimal codebooks from generalized bent ${\mathbb{Z}}_4$-valued quadratic forms

Codebooks with small maximum cross-correlations are desirable in many fields, such as compressed sensing, direct spread code-division multiple-access (CDMA) systems, and space-time codes. The objective of this paper is the construction of codebooks. Based on the theory of ${\mathbb{Z}}_4$-valued quadratic forms, we propose two classes of generalized bent functions over ${\mathbb{Z}}_4$, and construct new families of codebooks from these functions. The codebooks obtained in this paper are nearly optimal with respect to the Welch bound, and could have a very small alphabet size, which is of importance in practical applications. Moreover, some Boolean bent functions are also derived.