Algebraic constructions of graph-based nested codes from protographs

Nested codes have been employed in a large number of communication applications as a specific case of superposition codes, for example to implement binning schemes in the presence of noise, in joint network-channel coding, or in physical-layer secrecy. Whereas nested lattice codes have been proposed recently for continuous-input channels, in this paper we focus on the construction of nested linear codes for joint channel-network coding problems based on algebraic photograph LDPC codes. In particular, over the past few years several constructions of codes have been proposed that are based on random lifts of suitably chosen base graphs. More recently, an algebraic analog of this approach was introduced using the theory of voltage graphs. In this paper we illustrate how these methods can be used in the construction of nested codes from algebraic lifts of graphs.