  We study $k$-uniform states in heterogeneous systems whose local dimensions
are mixed. Based on the connections between mixed orthogonal arrays with
certain minimum Hamming distance, irredundant mixed orthogonal arrays and
$k$-uniform states, we present two constructions of $2$-uniform states in
heterogeneous systems. We also construct a family of $3$-uniform states in
heterogeneous systems, which solves a question posed in [D. Goyeneche et al.,
Phys. Rev. A 94, 012346 (2016)]. We also show two methods of generating
$(k-1)$-uniform states from $k$-uniform states. Some new results on the
existence and nonexistence of absolutely maximally entangled states are
provided. For the applications, we present an orthogonal basis consisting of
$k$-uniform states with minimum support. Moreover, we show that some
$k$-uniform bases can not be distinguished by local operations and classical
communications, and this shows quantum nonlocality with entanglement.
