  This letter develops an optimum beamforming method for downlink transmissions
in cell-free massive multiple-input multiple-output (MIMO) systems, which
employ a massive number of distributed access points to provide concurrent
services to multiple users. The optimum design is formulated as a max-min
problem that maximizes the minimum signal-to-interference-plus-noise ratio of
all users. It is shown analytically that the problem is quasi-concave, and the
optimum solution is obtained with the second-order cone programming. The
proposed method identifies the best achievable beamforming performance in
cell-free massive MIMO systems. The results can be used as benchmarks for the
design of practical low complexity beamformers.
