  In large-scale distributed storage systems, erasure codes are used to achieve
fault tolerance in the face of node failures. Tuning code parameters to
observed failure rates has been shown to significantly reduce storage cost.
Such tuning of redundancy requires "code conversion", i.e., a change in code
dimension and length on already encoded data. Convertible codes are a new class
of codes designed to perform such conversions efficiently. The access cost of
conversion is the number of nodes accessed during conversion.
  Existing literature has characterized the access cost of conversion of linear
MDS convertible codes only for a specific and small subset of parameters. In
this paper, we present lower bounds on the access cost of conversion of linear
MDS codes for all valid parameters. Furthermore, we show that these lower
bounds are tight by presenting an explicit construction for access-optimal
linear MDS convertible codes for all valid parameters. En route, we show that,
one of the degrees-of-freedom in the design of convertible codes that was
inconsequential in the previously studied parameter regimes, turns out to be
crucial when going beyond these regimes and adds to the challenge in the
analysis and code construction.
