Two classes of near-optimal codebooks with respect to the Welch bound

An \begin{document}$ (N,K) $\end{document} codebook \begin{document}$ {\mathcal C} $\end{document} is a collection of \begin{document}$ N $\end{document} unit norm vectors in a \begin{document}$ K $\end{document} -dimensional vectors space. In applications of codebooks such as CDMA, those vectors in a codebook should have a small maximum magnitude of inner products between any pair of distinct code vectors. In this paper, we propose two constructions of codebooks based on \begin{document}$ p $\end{document} -ary linear codes and on a hybrid character sum of a special kind of functions, respectively. With these constructions, two classes of codebooks asymptotically meeting the Welch bound are presented.