In this paper, we consider the use of lattice codes over Eisenstein integers for implementing a compute and-forward protocol in wireless networks when channel state information is not available at the transmitter. We extend the compute-and-forward paradigm of Nazer and Gastpar to decoding Eisenstein integer combinations of transmitted messages at relays by proving the existence of a sequence of pairs of nested lattices over Eisenstein integers in which the coarse lattice is good for covering and the fine lattice can achieve the Poltyrev limit. Using this result, we show that both the outage performance and error-correcting performance of the nested lattice codebooks over Eisenstein integers surpass those of lattice codebooks over integers considered by Nazer and Gastpar with no additional computational complexity.

We consider a class of lattices built using Construction A, where the underlying code is a non-binary spatially-coupled low density parity check code. We refer to these lattices as spatially-coupled LDA (SCLDA) lattices. SCLDA lattices can be constructed over integers, Gaussian integers and Eisenstein integers. We empirically study the performance of SCLDA lattices under belief propagation (BP) decoding. Ignoring the rate loss from termination, simulation results show that the BP thresholds of SCLDA lattices over integers is 0.11 dB (0.34 dB with the rate loss) and the BP thresholds for SCLDA lattices over Eisenstein integers are 0.08 dB from the Poltyrev limit (0.19 dB with the rate loss). Motivated by this result, we use SCLDA lattice codes over Eisenstein integers for implementing a compute-and-forward protocol. For the examples considered in this paper, the thresholds for the proposed lattice codes are within 0.28 dB from the achievable rate of this coding scheme and within 1.06 dB from the achievable computation rate of Nazer and Gastpar's coding scheme in [6] extended to Eisenstein integers.

We provide inner and outer bounds on the capacity region for the Gaussian bi-directional relaying over inter-symbol interference channels. The outer bound is obtained by the conventional cut-set argument. For the inner bound, we propose a compute-and-forward coding scheme based on lattice partition chains and study its achievable rate. The coding scheme is a time-domain coding scheme which uses a novel precoding scheme at the transmitter in combination with lattice precoding and a minimum mean squared error receiver to recover linear combinations of lattice codewords. The proposed compute-and-forward coding scheme substantially outperforms decode-and-forward schemes. While it is well known that for the point-to-point communication case, both independent coding along sub-channels and time-domain coding can approach the capacity limit, as a byproduct of the proposed scheme, we show that for the bi-directional relay case, independent coding along sub-channels is not optimal in general and joint coding across sub-channels can improve the capacity for some channel realizations.

We propose two compute-and-forward coding schemes for the bi-directional relay channel with inter- symbol interference (ISI) based on lattice codes and study their achievable rates. The first coding scheme is similar in spirit to coded orthogonal frequency division multiplexing (OFDM) with independent coding across sub-carriers and uses nested-lattice code with a power allocation strategy that can exploit the group property of lattices. The second coding scheme is a time-domain coding scheme which uses a novel precoding scheme at the transmitter in combination with lattice precoding and a minimum mean squared error receiver to recover linear combinations of lattice codewords. The proposed compute-and-forward coding schemes substantially outperform decode-and-forward schemes. While it is well known that for the point-to-point communication case, both the coded OFDM approach and the time-domain coding scheme can approach the capacity limit, we show that for the bi-directional relaying case, the performance of the two coding schemes are different. Particularly, we show that independent coding across sub-channels is not optimal and joint coding across sub-channels can improve the exchange capacity for some channel realizations.

We present a new coding scheme based on concatenating a newly introduced class of lattice codes called signal codes with interleaved Low Density Parity Check (LDPC)codes. These codes are shown to possess a special 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 recently proposed compute and forward paradigm. The decoding algorithm is based on an appropriate combination of the stack decoder with a message passing algorithm. Simulation results show that our proposed scheme can approach the uniform input AWGN capacity within 1.5 db, which is a 2 db improvement compared to using only signal codes when decoding using a stack algorithm with the same stack size. Simulation results for our proposed scheme applied to compute and forward are also presented.