  Exact-repair regenerating codes are considered for the case (n,k,d)=(4,3,3),
for which a complete characterization of the rate region is provided. This
characterization answers in the affirmative the open question whether there
exists a non-vanishing gap between the optimal bandwidth-storage tradeoff of
the functional-repair regenerating codes (i.e., the cut-set bound) and that of
the exact-repair regenerating codes. To obtain an explicit information
theoretic converse, a computer-aided proof (CAP) approach based on primal and
dual relation is developed. This CAP approach extends Yeung's linear
programming (LP) method, which was previously only used on information
theoretic problems with a few random variables due to the exponential growth of
the number of variables in the corresponding LP problem. The symmetry in the
exact-repair regenerating code problem allows an effective reduction of the
number of variables, and together with several other problem-specific
reductions, the LP problem is reduced to a manageable scale. For the
achievability, only one non-trivial corner point of the rate region needs to be
addressed in this case, for which an explicit binary code construction is
given.
