Abstract
In this paper, we give three constructions of codebooks with multiplicative characters of finite fields. Our constructions generalize the first construction in A. X. Zhang and K. Q. Feng (IEEE Trans. Inf. Theory 58(4), 2507-2511, 2012) from one dimension to two dimensions. We determine the maximum cross-correlation amplitude of these codebooks by the orthogonal relation of multiplicative characters and the properties of Jacobi sum. We prove that all the codebooks we constructed are asymptotically optimal with respect to the Welch bound. The parameters of these codebooks are new.
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