  Superposition codes are efficient for the Additive White Gaussian Noise
channel. We provide here a replica analysis of the performances of these codes
for large signals. We also consider a Bayesian Approximate Message Passing
decoder based on a belief-propagation approach, and discuss its performance
using the density evolution technic. Our main findings are 1) for the sizes we
can access, the message-passing decoder outperforms other decoders studied in
the literature 2) its performance is limited by a sharp phase transition and 3)
while these codes reach capacity as $B$ (a crucial parameter in the code)
increases, the performance of the message passing decoder worsen as the phase
transition goes to lower rates.
