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The purpose of this paper is to give explicit constructions of vectorialhyper-bent functions in the $\cP\cS_{ap}^{\#}$ class. It seems that theexplicit constructions were so far known only for very special cases. To thisend, we present a sufficient and necessary condition of this family ofvectorial functions to be hyper-bent. The conditions are expressed in terms ofgroup ring. Using this characterization, explicit constructions of vectorialhyper-bent functions of the $\cP\cS_{ap}^{\#}$ class via balanced functions areproposed. Furthermore, exact number of vectorial hyper-bent functions in the$\cP\cS_{ap}^{\#}$ class is found.  The results improve some previous work. Moreover, we solve a problem ofcounting vectorial hyper-bent functions left by Muratovi\'c-Ribi\'c, Pasalicand Ribi\'c in [{\em IEEE Trans. Inform. Theory}, 60 (2014), pp. 4408-4413].

On Constructions and Enumeration of Vectorial Hyper-bent Functions in the $\cP\cS_{ap}^{\#}$ Class