Asymptotic bounds on numbers of bent functions and partitions of the Boolean hypercube into linear and affine subspaces

The main result of the present paper is a new lower bound on the number of Boolean bent functions. This bound is based on a modification of the Maiorana--McFarland family of bent functions and recent progress in the estimation of the number of transversals in latin squares and hypercubes. In addition, we find the asymptotics of the logarithm of the numbers of partitions of the Boolean hypercube into $2$-dimensional linear and affine subspaces.