An algebraic framework for end-to-end physical-layer network coding

We propose an algebraic setup for end-to-end physical-layer network coding based on submodule transmission. We introduce a distance function between modules, describe how it relates to information loss and errors, and show how to compute it. Then we propose a definition of submodule error-correcting code, and investigate bounds and constructions for such codes. Introduction and motivation In the framework of physical-layer network coding (PNC) multiple terminals attempt to exchange messages through intermediate relays. The relays collect data from the terminals, and try to decode a function of the transmitted messages. Such function is then broadcasted to the terminals, which combine it with their side information to recover the other messages. In [11] the authors proposed a novel approach to PNC based on nested lattices, known as “compute-and-forward”. Under this approach, the structure of a fixed underlying lattice is exploited by the relays to decode the function of the messages, which is then forwarded to the terminals. As observed in [5], this communication scheme induces an end-to-end network coding channel with channel equation Y = AX + Z. (1) Here X is the transmitted matrix, whose rows are elements from a given ambient space Ω, A is a transfer matrix, and Z is an error matrix. In practice, A and Z are random matrices drawn according to certain distributions, that depend on the application at hand. A general algebraic framework to study and construct nested-lattice-based PNC schemes was recently proposed in [5] and further developed in [4]. Following this algebraic description, which is compatible with any underlying lattice, the message space Ω has the structure of a module over a principal ideal ring Ω = T/(d1)× T/(d2)× . . .× T/(dn), where T ⊆ C is a principal ideal domain (PID), d1, d2..., dn ∈ T are nonzero, non-invertible elements, and dn|dn−1| . . . |d1. Let R = T/(d1), then R is a principal ideal ring (PIR). The ambient space Ω is isomorphic to an R-submodule of Rn: Ω ∼= R× (d1/d2)× . . . × (d1/dn) ⊆ R . (2) E-mail addresses: [email protected], [email protected]. The authors were partially supported by the Swiss National Science Foundation through grant no. 200021 150207.