About the cubic part of the algebraic normal form of arbitrary bent functions

Maximally nonlinear Boolean functions in $n$ variables, where $n$ is even, are called bent functions. The algebraic normal form (ANF) is one of the most useful ways for representing Boolean functions. What can we say about ANF of bent functions? Is it true that linear, quadratic, cubic, etc. parts of bent functions can be arbitrary? Cases with linear and quadratic parts were studied previously. In this paper, we prove that cubic part of ANF of a bent function can not be arbitrary if $n=6, 8$.