Algebraic Approach to Physical-Layer Network Coding

The problem of designing physical-layer network coding (PNC) schemes via nested lattices is considered. Building on the compute-and-forward (C&F) relaying strategy of Nazer and Gastpar, who demonstrated its asymptotic gain using information-theoretic tools, an algebraic approach is taken to show its potential in practical, non-asymptotic, settings. A general framework is developed for studying nested-lattice-based PNC schemes---called lattice network coding (LNC) schemes for short---by making a direct connection between C&F and module theory. In particular, a generic LNC scheme is presented that makes no assumptions on the underlying nested lattice code. C&F is re-interpreted in this framework, and several generalized constructions of LNC schemes are given. The generic LNC scheme naturally leads to a linear network coding channel over modules, based on which non-coherent network coding can be achieved. Next, performance/complexity tradeoffs of LNC schemes are studied, with a particular focus on hypercube-shaped LNC schemes. The error probability of this class of LNC schemes is largely determined by the minimum inter-coset distances of the underlying nested lattice code. Several illustrative hypercube-shaped LNC schemes are designed based on Construction A and D, showing that nominal coding gains of 3 to 7.5 dB can be obtained with reasonable decoding complexity. Finally, the possibility of decoding multiple linear combinations is considered and related to the shortest independent vectors problem. A notion of dominant solutions is developed together with a suitable lattice-reduction-based algorithm.Comment: Submitted to IEEE Transactions on Information Theory, July 21, 2011. Revised version submitted Sept. 17, 2012. Final version submitted July 3, 201

Phase Precoded Compute-and-Forward with Partial Feedback

In this work, we propose phase precoding for the compute-and-forward (CoF) protocol. We derive the phase precoded computation rate and show that it is greater than the original computation rate of CoF protocol without precoder. To maximize the phase precoded computation rate, we need to 'jointly' find the optimum phase precoding matrix and the corresponding network equation coefficients. This is a mixed integer programming problem where the optimum precoders should be obtained at the transmitters and the network equation coefficients have to be computed at the relays. To solve this problem, we introduce phase precoded CoF with partial feedback. It is a quantized precoding system where the relay jointly computes both a quasi-optimal precoder from a finite codebook and the corresponding network equations. The index of the obtained phase precoder within the codebook will then be fedback to the transmitters. A "deep hole phase precoder" is presented as an example of such a scheme. We further simulate our scheme with a lattice code carved out of the Gosset lattice and show that significant coding gains can be obtained in terms of equation error performance.Comment: 5 Pages, 4 figures, submitted to ISIT 201

Lattices from Codes for Harnessing Interference: An Overview and Generalizations

In this paper, using compute-and-forward as an example, we provide an overview of constructions of lattices from codes that possess the right algebraic structures for harnessing interference. This includes Construction A, Construction D, and Construction $\pi_A$ (previously called product construction) recently proposed by the authors. We then discuss two generalizations where the first one is a general construction of lattices named Construction $\pi_D$ subsuming the above three constructions as special cases and the second one is to go beyond principal ideal domains and build lattices over algebraic integers