Lower bound for the number of bent functions at the minimum distance from Majorana — McFarland bent functions

The construction of bent functions at a certain distance from a given bent function is investigated. The criterion that the function obtained from the bent function $f$ by adding an indicator of an affine subspace of dimension $n$ is a bent function is proven, where $f$ belongs to the Maiorana — McFarland class $\mathcal{M}_{2n}$. It is shown that the lower bound $2^{2n+1} -2^n$ for the number of bent functions at the minimum distance from a bent function from the class $\mathcal{M}_{2n}$ is attained for prime $n \geq 5$. Bent functions are found for which the lower bound is attainable. It is shown that this lower bound is not attained for bent functions from the class $\mathcal{M}_{2n}$, where the permutation is not an APN function. For some distances, in particular $2^{2n-1}$, lower bounds for the number of bent functions in the class $\mathcal{M}_{2n}$ at these distances from bent functions in the class $\mathcal{C}$ are obtained.