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

Abstract

<p style='text-indent:20px;'>An <inline-formula><tex-math id="M1">\begin{document}$ (N,K) $\end{document}</tex-math></inline-formula> codebook <inline-formula><tex-math id="M2">\begin{document}$ {\mathcal C} $\end{document}</tex-math></inline-formula> is a collection of <inline-formula><tex-math id="M3">\begin{document}$ N $\end{document}</tex-math></inline-formula> unit norm vectors in a <inline-formula><tex-math id="M4">\begin{document}$ K $\end{document}</tex-math></inline-formula>-dimensional vectors space. In applications of codebooks such as CDMA, those vectors in a codebook should have a small maximum magnitude of inner products between any pair of distinct code vectors. In this paper, we propose two constructions of codebooks based on <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula>-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.