  Partial MDS (PMDS) and sector-disk (SD) codes are classes of erasure
correcting codes that combine locality with strong erasure correction
capabilities. We construct PMDS and SD codes with local regeneration where each
local code is a bandwidth-optimal regenerating MDS code. In the event of a node
failure, these codes reduce both, the number of servers that have to be
contacted as well as the amount of network traffic required for the repair
process. The constructions require significantly smaller field size than the
only other construction known in literature. Further, we present a construction
of PMDS codes with global regeneration which allow to efficiently repair
patterns of node failures that exceed the local erasure correction capability
of the code and thereby invoke repair across different local groups.
