  We consider the design of regenerating codes for distributed storage systems
at the minimum bandwidth regeneration (MBR) point. The codes allow for a repair
process that is exact and uncoded, but table-based. These codes were introduced
in prior work and consist of an outer MDS code followed by an inner fractional
repetition (FR) code where copies of the coded symbols are placed on the
storage nodes. The main challenge in this domain is the design of the inner FR
code.
  In our work, we consider generalizations of FR codes, by establishing their
connection with a family of combinatorial structures known as resolvable
designs. Our constructions based on affine geometries, Hadamard designs and
mutually orthogonal Latin squares allow the design of systems where a new node
can be exactly regenerated by downloading $\beta \geq 1$ packets from a subset
of the surviving nodes (prior work only considered the case of $\beta = 1$).
Our techniques allow the design of systems over a large range of parameters.
Specifically, the repetition degree of a symbol, which dictates the resilience
of the system can be varied over a large range in a simple manner. Moreover,
the actual table needed for the repair can also be implemented in a rather
straightforward way. Furthermore, we answer an open question posed in prior
work by demonstrating the existence of codes with parameters that are not
covered by Steiner systems.
