On the bent functions closest to a given Maiorana–McFarland bent function

Bent functions of $2n$ variables closest to a given bent function in the Maiorana–McFarland class are considered. The known criterion for their construction is revised and the method of calculating their number is refined. We investigate functions such that the number of closest bent functions is approximate to its lower and sharp upper bounds. The existence of bent functions whose number of closest bent functions has the same asymptotics as the lower bound is proven. Examples of functions in the Maiorana–McFarland class are given for which the calculated number of closest bent functions is close to the upper bound. Attainability of the lower bound is considered, and known necessary and sufficient conditions are refined. We show that the lower bound is attained for $n$ equaled to a power of a prime $p \geq 5$, as well as for some other $n.$ A complete classification of functions of $6$ variables in the Maiorana–McFarland class using the number of closest bent functions is obtained. Tab. 1, bibliogr. 40.