  Batch codes are a type of codes specifically designed for coded distributed
storage systems and private information retrieval protocols. These codes have
got much attention in recent years due to their ability to enable efficient and
secure storage in distributed systems.
  In this paper, we study an array code version of the batch codes, which is
called the \emph{batch array code} (BAC). Under the setting of BAC, each node
stores a bucket containing multiple code symbols and responds with a locally
computed linear combination of the symbols in its bucket during the recovery of
a requested symbol. We demonstrate that BACs can support the same type of
requests as the original batch codes but with reduced redundancy. Specifically,
we establish information theoretic lower bounds on the code lengths and provide
several code constructions that confirm the tightness of the lower bounds for
certain parameter regimes.
