  A unified graphical approach to random coding for any memoryless, single-hop,
K-user channel with or without common information is defined through two steps.
The first step is user virtualization: each user is divided into multiple
virtual sub-users according to a chosen rate-splitting strategy. This results
in an enhanced channel with a possibly larger number of users for which more
coding possibilities are available and for which common messages to any subset
of users can be encoded. Following user virtualization, the message of each
user in the enhanced model is coded using a chosen combination of coded
time-sharing, superposition coding and joint binning. A graph is used to
represent the chosen coding strategies: nodes in the graph represent codewords
while edges represent coding operations. This graph is used to construct a
graphical Markov model which illustrates the statistical dependency among
codewords that can be introduced by the superposition coding or joint binning.
Using this statistical representation of the overall codebook distribution, the
error probability of the code is shown to vanish via a unified analysis. The
rate bounds that define the achievable rate region are obtained by linking the
error analysis to the properties of the graphical Markov model. This proposed
framework makes it possible to numerically obtain an achievable rate region by
specifying a user virtualization strategy and describing a set of coding
operations. The union of these rate regions defines the maximum achievable rate
region of our unified coding strategy.
