Generalized bent functions with large dimension

Abstract

By a fundamental result by Mesnager et al. in 2018, a generalized bent function (originally defined as a class of functions from an $ n $-dimensional vector space $ \mathbb{V}_n^{(p)} $ into a cyclic group), is a bent function $ g: \mathbb{V}_n^{(p)}\rightarrow \mathbb{F}_p $ with a partition $ \mathcal{P} $ of $ \mathbb{V}_n^{(p)} $, such that for every function $ C $ which is constant on the sets of $ \mathcal{P} $, the function $ g + C $ is bent. The set of these bent functions forms then an affine space of dimension $ |\mathcal{P}| \le p^{n/2} $. This characterization of generalized bent functions is much more comprehensive than any earlier description.In this article, we analyse some classes of bent functions under this perspective. As shown earlier, Maiorana-McFarland bent functions permit the largest possible partitions, giving rise to $ p^{n/2} $-dimensional affine bent function spaces. The reason behind is their characterization as the bent functions, which are affine restricted to the $ n/2 $-dimensional affine subspaces of a trivial cover of $ \mathbb{V}_n^{(p)} $. We will show that maximal possible partitions can also be obtained for other classes of (regular) bent functions. Most notably, these classes are described as bent functions which are affine restricted to non-trivial covers of $ \mathbb{V}_n^{(p)} $. We investigate (largest) partitions for other types of bent functions, including weakly regular and non-weakly respectively non-dual bent functions. We round off the article giving partitions for bent functions obtained from bent partitions, which includes the partial spread class, and partitions for Carlet's class $ \mathcal{C} $ and $ \mathcal{D} $.