  This paper considers the Gaussian multiple-access channel (MAC) in the
asymptotic regime where the number of users grows linearly with the code
length. We propose efficient coding schemes based on random linear models with
approximate message passing (AMP) decoding and derive the asymptotic error rate
achieved for a given user density, user payload (in bits), and user energy. The
tradeoff between energy-per-bit and achievable user density (for a fixed user
payload and target error rate) is studied, and it is demonstrated that in the
large system limit, a spatially coupled coding scheme with AMP decoding
achieves near-optimal tradeoffs for a wide range of user densities.
Furthermore, in the regime where the user payload is large, we also study the
tradeoff between energy-per-bit and spectral efficiency and discuss methods to
reduce decoding complexity.
