On generalized bent and q-ary perfect nonlinear functions

The notion of bent function has been generalized by Kumar (1985) and other authors to the alphabet Z_q=Z/(qZ). The classical equivalent definitions of binary bent functions lead, through this generalization, to the notions of generalized bent functions and of q-ary perfect nonlinear functions. The first notion is weaker than the second one. We show that only one known construction of generalized bent functions can produce perfect nonlinear functions. It works for n even, (n>2). We introduce constructions of perfect nonlinear functions on Z ₄ⁿ, for every n>1