Six constructions of asymptotically optimal codebooks via the character sums

In this paper, using additive characters of finite field, we find a codebook which is equivalent to the measurement matrix in [20]. The advantage of our construction is that it can be generalized naturally to construct the other five classes of codebooks using additive and multiplicative characters of finite field. We determine the maximal cross-correlation amplitude of these codebooks by the properties of characters and character sums. We prove that all the codebooks we constructed are asymptotically optimal with respect to the Welch bound. The parameters of these codebooks are new.