  In 2017, Polyanskiy [1] showed that the trade-off between power and bandwidth
efficiency for massive Gaussian random access is governed by two fundamentally
different regimes: low power and high power. For both regimes, tight
performance bounds were found by Zadik et al. [2], in 2019. This work utilizes
recent results on the exact block error probability of Gaussian random codes in
additive white Gaussian noise to propose practical methods based on iterative
soft decoding to closely approach the bounds in [2]. In the low power regime,
this work finds that orthogonal random codes can be applied directly. In the
high power regime, a more sophisticated effort is needed. This work shows that
power-profile optimization by means of linear programming as pioneered by Caire
et al. [3], in 2001, is a promising strategy to apply. The proposed combination
of orthogonal random coding and iterative soft decoding even outperforms the
existence bounds of Zadik et al. [2] in the low power regime and is very close
to the non-existence bounds for message lengths around 100 and above. Finally,
the approach of power optimization by linear programming proposed for the high
power regime is found to benefit from power imbalances due to fading which
makes is even more attractive for typical mobile radio channels.
