On values of the affine rank of the support of spectrum of a plateaued function

We prove that the affine rank of any plateaued function with spectrum support of cardinality 16 is equal to 4, 5 or 6. For any positive integer $h$, we consider plateaued functions with spectrum support of cardinality $4^h$, give bounds for the affine rank of such functions and construct functions with affine rank equal to any possible value from $2h$ to $2^{h+1}-2$.