On tightness of the lower bound for the number of bent functions at the minimum distance from a bent function from the Maiorana —McFarland class

The lower bound $2^{2n+1} - 2^n$ for the number of bent functions at the minimum distance from a bent function from the Maiorana — McFarland class $\mathcal{M}_{2n}$ in $2n$ variables is investigated. A criterion for the reachability of this lower bound for functions in algebraic representation is presented. It is constructively proven that it is accurate for $n = p^k$, where $p \neq 2,3$ is prime and $k$ is natural. It is shown that a necessary condition for the reachability of the bound is the construction of a function from $\mathcal{M}_{2n}$ using an APN permutation whose set of values on any affine subspace of dimension $3$ is not an affine subspace.