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.