  We address the open problem of establishing the rate region for exact-repair
regenerating codes for given parameters (n,k,d). Tian determined the rate
region for a (4,3,3) code and found that it lies strictly within the
functional-repair rate region. Using different methods, Sasidharan, Senthoor
and Kumar proved a non-vanishing gap between the functional-repair outer bound
and the exact-repair outer bound for codes with k>=3. Our main results are two
improved outer bounds for exact-repair regenerating codes. They capture and
then extend essential parts in the proofs by Tian and by Sasidharan, Senthoor
and Kumar. We show that the bounds can be combined for further improvements.
