More constructions of near optimal codebooks associated with binary sequences

An $(N, K)$ codebook $\mathcal{C}$ is a collection of unit norm vectors in a $K$-dimensional vectors space. In applications of codebooks such as CDMA, those vectors in a codebook should have a small maximum magnitude of inner products, denoted by $I_{\max}(\mathcal{C})$, between any pair of distinct code vectors. Since the famous Welch bound is a lower bound on $I_{\max}(\mathcal{C})$, it is desired to construct codebooks meeting the Welch bound strictly or asymptotically. Recently, N. Y. Yu presents a method for constructing codebooks associated with a binary sequence from a $\Phi$-transform matrix. Using this method, he discovers new classes of codebooks with nontrivial bounds on the maximum inner products from Fourier and Hadamard matrices. We construct more near optimal codebooks by Yu's idea. We first provide more choices of binary sequences. We also show more choices of the $\Phi$-transform matrices. Therefore, we can present more codebooks $\mathcal{C}$ with nontrivial bounds on their $I_{\max}(\mathcal{C})$. Our work can serve as a complement of Yu's work.