  A noisy CDMA downlink channel operating under a strict complexity constraint
on the receiver is introduced. According to this constraint, detected bits,
obtained by performing hard decisions directly on the channel's matched filter
output, must be the same as the transmitted binary inputs. This channel
setting, allowing the use of the simplest receiver scheme, seems to be
worthless, making reliable communication at any rate impossible. However,
recently this communication paradigm was shown to yield valuable information
rates in the case of a noiseless channel. This finding calls for the
investigation of this attractive complexity-constrained transmission scheme for
the more practical noisy channel case. By adopting the statistical mechanics
notion of metastable states of the renowned Hopfield model, it is proved that
under a bounded noise assumption such complexity-constrained CDMA channel gives
rise to a non-trivial Shannon-theoretic capacity, rigorously analyzed and
corroborated using finite-size channel simulations. For unbounded noise the
channel's outage capacity is addressed and specifically described for the
popular additive white Gaussian noise.
