Communication over finite-ring matrix channels

Though network coding is traditionally performed over finite fields, recent work on nested-lattice-based network coding suggests that, by allowing network coding over finite rings, more efficient physical-layer network coding schemes can be constructed. This paper considers the problem of communication over a finite-chain-ring matrix channel Y = AX + BZ, where X is the channel input, Y is the channel output, Z is random noise, and A and B are random transfer matrices. Tight capacity results are obtained and simple polynomial-complexity capacity-achieving coding schemes are provided under certain distributions of A, B, and Z, extending the work of Silva, Kschischang and Kötter (2010), who handled the case of finite fields. This extension is based on several new results that generalize concepts and methods from matrices over finite fields to matrices over finite chain rings.