Replica analysis and approximate message passing decoder for superposition codes

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 technique. 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.