An Algebraic Framework for End-to-End Physical-Layer Network Coding
Elisa Gorla, Alberto Ravagnani · IEEE Transactions on Information Theory · 2017
We propose an algebraic framework for end-to-end physical-layer network coding based on submodules transmission. Our approach is motivated by nested-lattice-based network coding schemes, that naturally induce end-to-end channels where the ambient space has the structure of a module over a principal ideal ring. The setup is compatible with previously proposed approaches for finite chain rings, and extends them to arbitrary principal ideal rings. We introduce a distance function between modules, and describe how it relates to information loss and errors. We also show that computing the distance between modules reduces to computing the length of certain ideals in the base ring. We then propose a definition of submodule error-correcting code, and establish two upper bounds for the cardinality of these codes. Finally, we present some constructions of submodule codes, showing that they have asymptotically optimal cardinality for certain choices of the parameters.