  We consider the problem of efficient recovery of the data stored in any
individual node of a distributed storage system, from the rest of the nodes.
Applications include handling failures and degraded reads. We measure
efficiency in terms of the amount of data-read and the download required. To
minimize the download, we focus on the minimum bandwidth setting of the
'regenerating codes' model for distributed storage. Under this model, the
system has a total of n nodes, and the data stored in any node must be
(efficiently) recoverable from any d of the other (n-1) nodes. Lower bounds on
the two metrics under this model were derived previously; it has also been
shown that these bounds are achievable for the amount of data-read and download
when d=n-1, and for the amount of download alone when d<n-1.
  In this paper, we complete this picture by proving the converse result, that
when d<n-1, these lower bounds are strictly loose with respect to the amount of
read required. The proof is information-theoretic, and hence applies to
non-linear codes as well. We also show that under two (practical) relaxations
of the problem setting, these lower bounds can be met for both read and
download simultaneously.
