Constructions of bent functions on the minimal distance from the quadratic bent function

In this paper we study how to construct new bent functions by slight modifications of the initial one. The answer to this question is directly connected to the studying of bent functions on the minimal Hamming distance from the given bent function. Here we constructively describe all bent functions on the minimal distance from the quadratic bent function and calculate their exact number. We get a lower bound for the number of bent functions on the minimal distance from a bent function of Maiorana-McFarland type. We present several facts and hypotheses on the maximal number of bent functions that can be obtained in this way.