  We study the embedding $\text{id}: \ell_p^b(\ell_q^d) \to \ell_r^b(\ell_u^d)$
and prove matching bounds for the entropy numbers $e_k(\text{id})$ provided
that $0<p<r\leq \infty$ and $0<q\leq u\leq \infty$. Based on this finding, we
establish optimal dimension-free asymptotic rates for the entropy numbers of
embeddings of Besov and Triebel-Lizorkin spaces of small dominating mixed
smoothness which settles an open question in the literature. Both results rely
on a novel covering construction recently found by Edmunds and Netrusov.
