Asymptotic of Number of Similarity Classes of Commuting Tuples

We have for positive integers $n$, $k$ and finite field ${\mathbb{F}}_q$, $c(n,k,q)$, as the number of simultaneous similarity classes of $k$-tuples of commuting $n\times n$ matrices over the ${\mathbb{F}}_q$. In this paper, it has been shown that $c(n,k,q)$ as a function of $k$ for fixed $n$ and $q$ is asymptotically $q^{m(n)k}$, where $m(n)=\left[\frac{n^2}{4}\right]+1$, which is the dimension of the maximal commutative subalgebra of $M_n({\mathbb{F}}_q)$ (the algebra of $n\times n$ matrices over ${\mathbb{F}}_q$).