An upper bound for the number of bent functions at the distance $2^k$ from an arbitrary bent function in $2k$ variables

An upper bound for the number of bent functions at the distance $2^k$ from an arbitrary bent function in $2k$ variables is obtained. The bound is reached only for quadratic bent functions. The notion of completely affine decomposable Boolean function is introduced. It is proven that only affine and quadratic Boolean functions can be completely affine decomposable.