  We study the approximate message-passing decoder for sparse superposition
coding on the additive white Gaussian noise channel and extend our preliminary
work [1]. We use heuristic statistical-physics-based tools such as the cavity
and the replica methods for the statistical analysis of the scheme. While
superposition codes asymptotically reach the Shannon capacity, we show that our
iterative decoder is limited by a phase transition similar to the one that
happens in Low Density Parity check codes. We consider two solutions to this
problem, that both allow to reach the Shannon capacity: i) a power allocation
strategy and ii) the use of spatial coupling, a novelty for these codes that
appears to be promising. We present in particular simulations suggesting that
spatial coupling is more robust and allows for better reconstruction at finite
code lengths. Finally, we show empirically that the use of a fast
Hadamard-based operator allows for an efficient reconstruction, both in terms
of computational time and memory, and the ability to deal with very large
messages.
