Let L be a finite relational language and alpha = (alpha(R): R is an element of L) a tuple with 0 < alpha(R) <= 1 for each R is an element of L. Consider a dimension function
delta(alpha) (A) = vertical bar A vertical bar - Sigma(R is an element of L) alpha(R)e(R)(A)
where each e(R)(A) is the number of realizations of R in A. Let K(alpha) be the class of finite structures A such that delta(alpha)(X) >= 0 for my substructure X of A. We show that the theory of the generic model of K(alpha) is AE-axiomatizable for any alpha.