  This paper studies two crucial problems in the context of coded distributed
storage systems directly related to their performance: 1) for a fixed alphabet
size, determine the minimum number of servers the system must have for its
service rate region to contain a prescribed set of points; 2) for a given
number of servers, determine the minimum alphabet size for which the service
rate region of the system contains a prescribed set of points. The paper
establishes rigorous upper and lower bounds, as well as code constructions
based on techniques from coding theory, optimization, and projective geometry.
