Generalized bent functions with large dimension

Official URL: https://dx.doi.org/10.3934/amc.2023004

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 V(p) n a partition P of V(p) n into a cyclic group), is a bent function g: Vn(p) → Fp with, such that for every function C which is constant on the sets of P, the function g +C is bent. The set of these bent functions forms then an affine space of dimension |P| ≤ pn/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 perspec-tive. As shown earlier, Maiorana-McFarland bent functions permit the largest possible partitions, giving rise to pn/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 V(p) n. We will show that maximal possible partitions can also be obtained for other classes of (regular) bent functions. Most notably, these classes are de-scribed as bent functions which are affine restricted to non-trivial covers of V(p) n. We investigate (largest) partitions for other types of bent functions, in-cluding 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 C and D.