  Partial maximum distance separable (PMDS) codes are a kind of erasure codes
where the nodes are divided into multiple groups with each forming an MDS code
with a smaller code length, thus they allow repairing a failed node with only a
few helper nodes and can correct all erasure patterns that are
information-theoretically correctable. However, the repair of a failed node of
PMDS codes still requires a large amount of communication if the group size is
large. Recently, PMDS array codes with each local code being an MSR code were
introduced to reduce the repair bandwidth further. However, they require
extensive rebuilding access and unavoidably a significant sub packetization
level. In this paper, we first propose two constructions of PMDS array codes
with two global parities that have smaller sub-packetization levels and much
smaller finite fields than the existing one. One construction can support an
arbitrary number of local parities and has $(1+\epsilon)$-optimal repair
bandwidth (i.e., $(1+\epsilon)$ times the optimal repair bandwidth), while the
other one is limited to two local parities but has significantly smaller
rebuilding access and its sub packetization level is only $2$. In addition, we
present a construction of PMDS array code with three global parities, which has
a smaller sub-packetization level as well as $(1+\epsilon)$-optimal repair
bandwidth, the required finite field is significantly smaller than existing
ones.
