Hybrid Character Sums From Vectorial Dual-Bent Functions and Asymptotically Optimal Complex Codebooks With Small Alphabet Sizes

Hybrid character sums are an important class of exponential sums which have nice applications in coding theory and sequence design. Let Fpmbe the finite field withpmelements for a primepand a positive integerm. LetV(p)nbe ann-dimensional vector space over Fpfor a primep. In this paper, we study the hybrid character sums of the form Σx∈V(p)nψ (F(x)) χa(x), whereFis a function fromV(p)nto Fpm,a∈V(p)n, ψ is a nontrivial multiplicative character of Fpm, χa(x) = ζ⟨a,x⟩npis the character ofV(p)nand ⟨, ⟩ndenotes a (non-degenerate) inner product ofV(p)n. IfF(x) is a vectorial dual-bent function anda∈V(p)n\ {0}, we determine their complex modulus or explicit values under certain conditions. This generalizes some known results as special cases. We show that the hybrid character sums from vectorial dual-bent functions have very small complex modulus. As applications, three families of asymptotically optimal complex codebooks are constructed from vectorial dual-bent functions and their maximal cross-correlation amplitude are determined based on the hybrid character sums. The codebooks we construct have very small alphabet sizes. This enhances their appeal for implementation. Besides, all of the three families of codebooks have only two-valued or three-valued cross-correlation amplitudes.