Lattice-Based Coding Schemes for Wireless Relay Networks

Compute-and-forward is a novel relaying paradigm in wireless communications in which relays in a network directly compute or decode functions of signals transmitted from multiple transmitters and forward them to a central destination. In this dissertation, we study three problems related to compute-and-forward. In the first problem, we consider the use of lattice codes for implementing a compute-and-forward protocol in wireless networks when channel state information is not available at the transmitter. We propose the use of lattice codes over Eisenstein integers and we prove the existence of a sequence of lattices over Eisenstein integers which are good for quantization and achieve capacity over an additive white Gaussian noise (AWGN) channel. Using this, we show that the information rates achievable with nested lattice codebooks over Eisenstein integers are higher than those achievable with nested lattice codebooks over integers considered by Nazer and Gastpar in [6] in the average sense. We also propose a separation-based framework for compute-and-forward that is based on the concatenation of a non-binary linear code with a modulation scheme derived from the ring of Eisenstein integers, which enables the coding gain and shaping gain to be separated, resulting in significantly higher theoretically achievable computation rates. In the second problem, we construct lattices based on spatially-coupled low-density parity check (LDPC) codes and empirically show that such lattices can approach the Poltyrev limit very closely for the point-to-point unconstrained AWGN channel. We then employ these lattices to implement a compute-and-forward protocol and empirically show that these lattices can approach the theoretically achievable rates closely. In the third problem, we present a new coding scheme based on concatenating a newly introduced class of lattice codes called convolutional lattice codes with LDPC codes, which we refer to as concatenated convolutional lattice codes (CCLS) and study their application to compute-and-forward (CF). The decoding algorithm for CCLC is based on an appropriate combination of the stack decoder with a message passing algorithm, and is computationally much more efficient than the conventional decoding algorithm for convolutional lattice codes. Simulation results show that CCLC can approach the point-to-point uniform input AWGN capacity very closely with soft decision decoding. Also, we show that they possess the required algebraic structure which makes them suitable for recovering linear combinations (over a finite field) of the transmitted signals in a multiple access channel. This facilitates their use as a coding scheme for the compute-and-forward paradigm. Simulation results show that CCLC can approach theoretically achievable rates very closely when implemented for the compute-and-forward

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

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

Efficient Integer Coefficient Search for Compute-and-Forward

Integer coefficient selection is an important decoding step in the implementation of compute-and-forward (C-F) relaying scheme. Choosing the optimal integer coefficients in C-F has been shown to be a shortest vector problem (SVP) which is known to be NP hard in its general form. Exhaustive search of the integer coefficients is only feasible in complexity for small number of users while approximation algorithms such as Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm only find a vector within an exponential factor of the shortest vector. An optimal deterministic algorithm was proposed for C-F by Sahraei and Gastpar specifically for the real valued channel case. In this paper, we adapt their idea to the complex valued channel and propose an efficient search algorithm to find the optimal integer coefficient vectors over the ring of Gaussian integers and the ring of Eisenstein integers. A second algorithm is then proposed that generalises our search algorithm to the Integer-Forcing MIMO C-F receiver. Performance and efficiency of the proposed algorithms are evaluated through simulations and theoretical analysis.Comment: IEEE Transactions on Wireless Communications, to appear.12 pages, 8 figure