Communication over Finite-Chain-Ring Matrix Channels

TL;DR

This paper extends finite field network coding results to finite-chain-ring matrix channels, deriving capacity and coding schemes for more efficient physical-layer network coding over rings.

Contribution

It introduces capacity results and polynomial-complexity coding schemes for finite-ring matrix channels, generalizing finite field methods to chain rings.

Findings

Capacity results for finite-ring matrix channels

Polynomial-complexity capacity-achieving codes

Extension of finite field matrix concepts to chain rings

Abstract

Though network coding is traditionally performed over finite fields, recent work on nested-lattice-based network coding suggests that, by allowing network coding over certain finite rings, more efficient physical-layer network coding schemes can be constructed. This paper considers the problem of communication over a finite-ring matrix channel $Y = AX + BE$, where $X$ is the channel input, $Y$ is the channel output, $E$ is random error, and $A$ and $B$ are random transfer matrices. Tight capacity results are obtained and simple polynomial-complexity capacity-achieving coding schemes are provided under the assumption that $A$ is uniform over all full-rank matrices and $BE$ is uniform over all rank-$t$ matrices, extending the work of Silva, Kschischang and K\"{o}tter (2010), who handled the case of finite fields. This extension is based on several new results, which may be of independent interest, that generalize concepts and methods from matrices over finite fields to matrices over finite chain rings.