  Network-coded multiple access (NCMA) is a communication scheme for wireless
multiple-access networks where physical-layer network coding (PNC) is employed.
In NCMA, a user encodes and spreads its message into multiple packets. Time is
slotted and multiple users transmit packets (one packet each) simultaneously in
each timeslot. A sink node aims to decode the messages of all the users from
the sequence of receptions over successive timeslots. For each timeslot, the
NCMA receiver recovers multiple linear combinations of the packets transmitted
in that timeslot, forming a system of linear equations. Different systems of
linear equations are recovered in different timeslots. A message decoder then
recovers the original messages of all the users by jointly solving multiple
systems of linear equations obtained over different timeslots. We propose a
low-complexity digital fountain approach for this coding problem, where each
source node encodes its message into a sequence of packets using a fountain
code. The aforementioned systems of linear equations recovered by the NCMA
receiver effectively couple these fountain codes together. We refer to the
coupling of the fountain codes as a linearly-coupled (LC) fountain code. The
ordinary belief propagation (BP) decoding algorithm for conventional fountain
codes is not optimal for LC fountain codes. We propose a batched BP decoding
algorithm and analyze the convergence of the algorithm for general LC fountain
codes. We demonstrate how to optimize the degree distributions and show by
numerical results that the achievable rate region is nearly optimal. Our
approach significantly reduces the decoding complexity compared with the
previous NCMA schemes based on Reed-Solomon codes and random linear codes, and
hence has the potential to increase throughput and decrease delay in
computation-limited NCMA systems.
