In this paper, we analyze and derive conditions for stability of a feedback system in which the plant and feedback controller are separated by a noiseless finite-rate communication channel. We allow for two deterministic classes of reference inputs to excite the system, and derive sufficient conditions for input-output (IO) stability as a function of the encoding strategy and controller. We first construct an encoder as a quantizer that can have infinite memory and can be time-varying, in that the strategy it follows to allocate a total of R bits to its inputs, is a function of time. This construction of the quantizer leads to the result that the set of allocation strategies that maintains stability for each class of reference signals is convex, allowing the search for the most efficient strategy to ensure stability to be formulated as a convex optimization problem. We then synthesize quantizers and time-varying controllers to minimize the rate required for stability and to track commands. Examples presented in this paper demonstrate how this framework enables computationally efficient methods for simultaneously designing quantizers and controllers for given plants. Furthermore, we observe that our finite memory quantizers that minimize the rate required for stability do not reduce to trivial memoryless bit-allocation strategies.