An Algebraic Approach to Physical-Layer Network Coding

Abstract

The problem of designing new physical-layer network coding (PNC) schemes via lattice partitions is considered. Building on a recent work by Nazer and Gastpar, who demonstrated its asymptotic gain using information-theoretic tools, we take an algebraic approach to show its potential in non-asymptotic settings. We first relate Nazer-Gastpar's approach to the fundamental theorem of finitely generated modules over a principle ideal domain. Based on this connection, we generalize their code construction and simplify their encoding and decoding methods. This not only provides a transparent understanding of their approach, but more importantly, it opens up the opportunity to design efficient and practical PNC schemes. Finally, we apply our framework for PNC to a Gaussian relay network and demonstrate its advantage over conventional PNC schemes.Comment: 5 pages, 3 figures, accepted to IEEE Int. Symp. Information Theory, 201

On Multiplicative Matrix Channels over Finite Chain Rings

“…. , s 6 be the physical signals (complex vectors coming from a given lattice [7], [8]) transmitted by the nodes 1, 2, . .…”

“…The motivation comes from physical-layer network coding [6]. Indeed, the results in [7] show that the modulation employed at the physical layer induces a "matched choice" for the ring to be used in the linear network coding layer. For instance (see [7]), if uncoded quaternary phase-shift keying (QPSK) is employed, then the underlying ring should be chosen as R = Z 2 [i] = {0, 1, i, 1 + i}, which is not a finite field, but a finite chain ring.…”

“…For instance (see [7]), if uncoded quaternary phase-shift keying (QPSK) is employed, then the underlying ring should be chosen as R = Z 2 [i] = {0, 1, i, 1 + i}, which is not a finite field, but a finite chain ring. More generally, this is also true for wireless networks employing compute-and-forward [8] over arbitrary nested lattices: in this case, the underlying ring happens to be a principal ideal domain T (typically the integers, Z, the Gaussian integers, Z[i], or the Eisenstein integers, Z[ω]), with the corresponding message space W being a finite Tmodule [7]. As such,…”

Multihop compute‐and‐forward for generalised two‐way relay channels

“…The message rate of the single-access transmission event is equivalent to the achievable rate of point-to-point transmission given in Equation (19). Thus, the outage probability of the single-access transmission event f j S;l can be given by…”

“…The maximisation problem of the computation rate was converted to the shortest vector problem (SVP) in [17,18]. To overcome the difficulty of finding a good lattice partition for CaF, we took an algebraic approach in [19] to study a class of CaF schemes on the basis of practical lattice partitions. In [18], Feng et al generalised the framework by Nazer and Gastpar and introduced lattice network coding.Although there has been some recent research on CaF and its implementations, to the best of our knowledge, there has been no published work on the analysis of the outage performance of CaF in the MH-TRC.…”