  The max-flow outer bound is achievable by regenerating codes for functional
repair distributed storage system. However, the capacity of exact repair
distributed storage system is an open problem. In this paper, the linear
programming bound for exact repair distributed storage systems is formulated. A
notion of symmetrical sets for a set of random variables is given and
equalities of joint entropies for certain subsets of random variables in a
symmetrical set is established. Concatenation coding scheme for exact repair
distributed storage systems is proposed and it is shown that concatenation
coding scheme is sufficient to achieve any admissible rate for any exact repair
distributed storage system. Equalities of certain joint entropies of random
variables induced by concatenation scheme is shown. These equalities of joint
entropies are new tools to simplify the linear programming bound and to obtain
stronger converse results for exact repair distributed storage systems.
