Asymptotic of number of similarity classes of commuting tuples

Let c(n, k, q) be the number of simultaneous similarity classes of k-tuples of commuting n x n matrices over a finite field of order q. We show that, for a fixed n and q, c(n, k, q) is asymptotically q(m(n)k) (upto some constant factor), as a function of k, where m(n) = [n(2)/4] + 1 is the maximal dimension of a commutative subalgebra of the algebra of n x n matrices over the finite field.