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. A notion of completely affine decomposable Boolean function is introduced. It is proved that only affine and quadratic Boolean functions can be completely affine decomposable.