  We consider the design of multiuser constellations for a multiple access
channel (MAC) with K users, with M antennas each, that transmit simultaneously
to a receiver equipped with N antennas through a Rayleigh block-fading channel,
when no channel state information (CSI) is available to either the transmitter
or the receiver. In full-diversity scenarios where the coherence time is at
least T>= (K+1)M, the proposed constellation design criterion is based on the
asymptotic expression of the multiuser pairwise error probability (PEP) derived
by Brehler and Varanasi. In non-full diversity scenarios, for which the
previous PEP expression is no longer valid, the proposed design criteria are
based on proxies of the PEP recently proposed by Ngo and Yang. Although both
the PEP expression and its bounds or proxies were previously considered
intractable for optimization, in this work we derive their respective
unconstrained gradients. These gradients are in turn used in the optimization
of the proposed cost functions in different Riemannian manifolds representing
different power constraints. In particular, in addition to the standard unitary
space-time modulation (USTM) leading to optimization on the Grassmann manifold,
we consider a more relaxed per-codeword power constraint leading to
optimization on the so-called oblique manifold, and an average power constraint
leading to optimization on the so-called trace manifold. Equipped with these
theoretical tools, we design multiuser constellations for the MIMO MAC in
full-diversity and non-full-diversity scenarios with state-of-the-art
performance in terms of symbol error rate (SER).
