  We propose efficient and low-complexity multiuser detection (MUD) algorithms
for Gaussian multiple access channel (G-MAC) for short-packet transmission in
massive machine type communications. To do so, we first formulate the G-MAC MUD
problem as a sparse signal recovery problem and obtain the exact and
approximate joint prior distribution of the sparse vector to be recovered.
Then, we employ the Bayesian approximate message passing (AMP) algorithms with
the optimal separable and non-separable minimum mean squared error (MMSE)
denoisers for soft decoding of the sparse vector. The effectiveness of the
proposed MUD algorithms for a large number of devices is supported by
simulation results. For packets of 8 information bits, while the
state-of-the-art AMP with soft-threshold denoising achieves 8/100 of the upper
bound at Eb/N0 = 4 dB, the proposed algorithms reach 4/7 and 1/2 of the upper
bound.
