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×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)=[n²/4]+1 is the maximal dimension of a commutative subalgebra of the algebra of n×n matrices over the finite field.