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This paper focuses on the design of non-orthogonal multiple access (NOMA) ina classical two-transmitter two-receiver Z-channel, wherein one transmittersends information to its intended receiver from the direct link while the othertransmitter sends information to both receivers from the direct and crosslinks. Unlike most existing designs using (continuous) Gaussian inputdistribution, we consider the practical finite-alphabet (i.e., discrete) inputsby assuming that the widely-used quadrature amplitude modulation (QAM)constellations are adopted by both transmitters. To balance the errorperformance of two receivers, we apply the max-min fairness design criterion inthis paper. More specifically, we propose to jointly optimize the scalingfactors at both transmitters, which control the minimum Euclidean distance oftransmitting constellations, to maximize the smaller minimum Euclidean distanceof two resulting constellations at the receivers, subject to an individualaverage power constraint at each transmitter. The formulated problem is a mixedcontinuous-discrete optimization problem and is thus intractable in general. Byresorting to the Farey sequence, we manage to attain the closed-form expressionfor the optimal solution to the formulated problem. This is achieved bydividing the overall feasible region of the original optimization problem intoa finite number of sub-intervals and deriving the optimal solution in eachsub-interval. Through carefully observing the structure of the optimalsolutions in all sub-intervals, we obtain compact and closed-form expressionsfor the optimal solutions to the original problem in three possible scenariosdefined by the relative strength of the cross link. Simulation studies areprovided to validate our analysis and demonstrate the merits of the proposeddesign over existing orthogonal or non-orthogonal schemes.

On Non-Orthogonal Multiple Access with Finite-Alphabet Inputs in Z-Channels