  We study the problem of centralized exact repair of multiple failures in
distributed storage. We describe constructions that achieve a new set of
interior points under exact repair. The constructions build upon the layered
code construction by Tian et al., designed for exact repair of single failure.
We firstly improve upon the layered construction for general system parameters.
Then, we extend the improved construction to support the repair of multiple
failures, with varying number of helpers. In particular, we prove the
optimality of one point on the functional repair tradeoff of multiple failures
for some parameters. Finally, considering minimum bandwidth cooperative repair
(MBCR) codes as centralized repair codes, we determine explicitly the best
achievable region obtained by space-sharing among all known points, including
the MBCR point.
