  When a node in a distributed storage system fails, it needs to be promptly
repaired to maintain system integrity. While typical erasure codes can provide
a significant storage advantage over replication, they suffer from poor repair
efficiency. Locally repairable codes (LRCs) tackle this issue by reducing the
number of nodes participating in the repair process (locality), at the cost of
reduced minimum distance. In this paper, we study the tradeoff between locality
and minimum distance of LRCs with local codes that have arbitrary distance
requirements. Unlike existing methods where both the locality and the local
distance requirements imposed on every node are identical, we allow the
requirements to vary arbitrarily from node to node. Such a property can be an
advantage for distributed storage systems with non-homogeneous characteristics.
We present Singleton-type distance upper bounds and also provide an optimal
code construction with respect to these bounds. In addition, the feasible rate
region is characterized by dimension upper bounds that do not depend on the
distance.
