Characterizations of Bent and Almost Bent Function on ${\mathbb{Z}_p^2}$

Abstract

Bent and almost-bent functions on ${\mathbb{Z}_p^2}$ are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566–582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on ${\mathbb{Z}_p^2}$ are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on ${\mathbb{Z}_p^2}$ in two classes of ${\mathcal{M}}$ ’s and ${\mathcal{PS}}$ ’s, and show that the graph set corresponding to a bent function on ${\mathbb{Z}_p^2}$ can be written as the sum of a graph set of ${\mathcal{M}}$ ’s type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of ${\mathcal{M}}$ ’s type if its corresponding set contains more than (p − 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505–525, 1995) is therefore partially answered.