Generalized bent functions and their properties

Jet J_q^m denote the set of m-tuples over the integers modulo q and set $\text{i=}\text{?1}$, $\text{w = e}{}^{\mathrm{<mtext>i(</mtext><mtext>2\pi </mtext><mtext>q</mtext><mtext>)</mtext>}}$. As an extension of Rothaus' notion of a bent function, a function f, f: J_q^m → J_q¹ is called bent if all the Fourier coefficients of w^f have unit magnitude. An important feature of these functions is that their out-of-phase autocorrelation value is identically zero. The nature of the Fourier coefficients of a bent function is examined and a proof for the non-existence of bent functions over J_q^m, m odd, is given for many values of q of the form q = 2 (mod 4). For every possible value of q and m (other than m odd and q = 2 (mod 4)), constructions of bent functions are provided.