A new kind of hybrid character sums and their applications

In this paper, we introduce a new kind of hybrid character sums and their applications in the constructions of Mutually unbiased bases (MUBs) and codebooks. It is well-known that MUBs have significant applications in quantum information processing and other fields. However, only a few constructions of MUBs are known in the literature. In this note, we provide a recursive constructions of approximately mutually unbiased bases. Meanwhile, we present some constructions of asymptotically optimal codebooks by using the hybrid character sums.