Generalized bent functions into $\mathbb {Z}_{p^{k}}$ from the partial spread and the Maiorana-McFarland class

Abstract

Functions f from ${\mathbb {F}_{p}^{n}}$, n = 2m, to $\mathbb {Z}_{{p}^{k}}$ for which the character sum $\mathcal {H}^{k}_{f}(p^{t},u)=\sum\limits _{x\in {\mathbb {F}_{p}^{n}}}\zeta _{p^{k}}^{p^{t}f(x)}\zeta _{p}^{u\cdot x}$ (where $\zeta _{q} = e^{2\pi i/q}$ is a q-th root of unity), has absolute value $p^{m}$ for all $u\in {\mathbb {F}_{p}^{n}}$ and $0\le t\le k-1$, induce relative difference sets in ${\mathbb {F}_{p}^{n}}\times \mathbb {Z}_{{p}^{k}}$ hence are called bent. Functions only necessarily satisfying $|\mathcal {H}^{k}_{f}(1,u)| = p^{m}$ are called generalized bent. We show that with spreads we not only can construct a variety of bent and generalized bent functions, but also can design functions from ${\mathbb {F}_{p}^{n}}$ to $\mathbb {Z}_{{p}^{m}}$ satisfying $|\mathcal {H}_{f}^{m}(p^{t},u)| = p^{m}$ if and only if $t\in T$ for any $T\subset \{0,1\ldots ,m-1\}$. A generalized bent function can also be seen as a Boolean (p-ary) bent function together with a partition of ${\mathbb {F}_{p}^{n}}$ with certain properties. We show that the functions from the completed Maiorana-McFarland class are bent functions, which allow the largest possible partitions.