On the Duals of Generalized Bent Functions

In this paper, we study the duals of generalized bent functions f: V_{n}\rightarrow \mathbb {Z}_{p^{k}} , where V_{n} is an n -dimensional vector space over \mathbb {F}_{p} and p is an odd prime, k is a positive integer. It is known that weakly regular generalized bent functions always appear in pairs since the dual of a weakly regular generalized bent function is also a weakly regular generalized bent function. The duals of non-weakly regular generalized bent functions can be generalized bent or not generalized bent. By generalizing the construction of Çeşmelioğlu et al. , 2016, we provide an explicit construction of generalized bent functions whose duals can be generalized bent or not generalized bent. We show that the well-known direct sum construction and the generalized indirect sum construction given in Wang and Fu, 2021. can provide secondary constructions of generalized bent functions whose duals can be generalized bent or not generalized bent. By using the knowledge on ideal decomposition in cyclotomic fields, we prove that f^{**}(x)=f(-x) if f is a generalized bent function and its dual f^{*} is also a generalized bent function. For any non-weakly regular generalized bent function f which satisfies that f(x)=f(-x) and its dual f^{*} is generalized bent, we give a property and as a consequence, we prove that there is no self-dual generalized bent function f: V_{n}\rightarrow \mathbb {Z}_{p^{k}} if p\equiv 3 ~(mod ~4) and