  We construct a class of metric spaces whose transfinite asymptotic dimension
and complementary-finite asymptotic dimension are both $\omega+k$ for any
$k\in\mathbb{N}$, where $\omega$ is the smallest infinite ordinal number and a
metric space whose transfinite asymptotic dimension and complementary-finite
asymptotic dimension are both $2\omega$. Moreover, we study the relationship
between asymptotic dimension growth, transfinite asymptotic dimension and
finite decomposition complexity.
