  Lattices possess elegant mathematical properties which have been previously
used in the literature to show that structured codes can be efficient in a
variety of communication scenarios, including coding for the additive white
Gaussian noise (AWGN) channel, dirty-paper channel, Wyner-Ziv coding, coding
for relay networks and so forth. We consider the family of single-transmitter
multiple-receiver Gaussian channels where the source transmits a set of common
messages to all the receivers (multicast scenario), and each receiver has
'coded side information', i.e., prior information in the form of linear
combinations of the messages. This channel model is motivated by applications
to multi-terminal networks where the nodes may have access to coded versions of
the messages from previous signal hops or through orthogonal channels. The
capacity of this channel is known and follows from the work of Tuncel (2006),
which is based on random coding arguments. In this paper, following the
approach of Erez and Zamir, we design lattice codes for this family of channels
when the source messages are symbols from a finite field 'Fp' of prime size.
Our coding scheme utilizes Construction A lattices designed over the same prime
field 'Fp', and uses algebraic binning at the decoders to expurgate the channel
code and obtain good lattice subcodes, for every possible set of linear
combinations available as side information. The achievable rate of our coding
scheme is a function of the size 'p' of underlying prime field, and approaches
the capacity as 'p' tends to infinity.
