I Introduction
The interference channel [HanKobayashi1981] has been considered as a canonical model to understand the design principles of cellular networks with intercell interference. For example, the HanKobayashi scheme [HanKobayashi1981, EtkinTseWang2008] has been developed for the interference channel and is currently the best known strategy for the twouser interference channel. The HanKobayashi scheme operates by superposition encoding and sending two code components, a private part and a common part. Both components are optimally recovered by using simultaneous joint decoding [ChongMotaniGargElGamal2008, BandemerElGamalKim2012]. While the theoretical aspects of this strategy is well understood, how to practically implement it with low complexity remains as an important question. Since recovering the private part can be understood as a simple application of pointtopoint codes, the main challenge is to send the common messages, i.e., each receiver treats the interference channel as a multiple access channel and recovers both messages (decoding interference). The goal of this work is to design practical lowcomplexity decoders that can closely attain the promised performance of simultaneous decoding over multiple access channels.
Compared to the multiple access channel, the major difficulty in designing practical strategies for decoding interference in an interference channel is the following. In a multiple access channel, it is well known that sequential decoding (e.g. successive cancellation) and time sharing or ratesplitting can achieve the capacity region of the multiple–access channel [RimoldiUrbanke1996, GrantRimoldiUrbankeWhiting2001, ElGamalKim2011]. However, this is not the case when we treat the interference channel as a combination of two MACs (compound MAC), that is, a channel in which the pair of encoders send messages to both decoders. It was well observed in [FawziSavov2012, WangSasogluKim2014] that successive cancellation decoding with time sharing is strictly suboptimal compared to joint decoding for this case.
Recently, there has been some advances in designing lowcomplexity sequential decoding strategies that can overcome this issue. In particular, Wang, Şaşoğlu, and Kim [WangSasogluKim2014] presented the slidingwindow superposition coding (SWSC) strategy that achieves the performance of simultaneous decoding with existing pointtopoint codes. A case study using standard LTE codes of the SWSC scheme was given in [ParkKimWang2014, KimAhnKimParkWangChenPark2015]. An important component of this strategy is that they use block Markov encoding which requires multiblock transmissions. In another line of work, Zhu and Gastpar [ZhuGastpar2014b] proposed the compute–forward multiple access (CFMA) strategy based on the compute–and–forward framework proposed in [NazerGastpar2011]. It is shown that CFMA achieves the capacity region of the Gaussian MAC with sequential decoding when the SNR of both decoders is greater than , and the strategy is also extended to the interference channel in [zhu_lattice_2015]. Under this condition, the optimal performance is achieved using single block transmissions. The main component of CFMA is that it utilizes lattice codes to first compute a linear combination of the codewords sent by the transmitters, which is accomplished by extending the compute–and–forward strategy originally proposed by Nazer and Gastpar [NazerGastpar2011]. In the next stage, by using the linear combination of the codewords as side information, the decoder recovers any one of the messages^{1}^{1}1In general the CFMA strategy finds and computes the best two linearly independent combinations.. By having one linear combination and one of the other messages, the receiver can recover both messages. By appropriately scaling the lattice codes, it is shown in [ZhuGastpar2014b] that the dominant face of the capacity region of the multiple access channel can be attained with sequential decoders without time sharing, and thus, is optimal for the compound MAC setting under the SNR condition. Following the theoretical study of CFMA [ZhuGastpar2014b] which was attained with lattice codes and lattice decoding [ErezZamir2004], the main goal of our work is to design practical codes and efficient decoding algorithms that can attain the achievable rates of CFMA. To accomplish this goal and for applications to current systems, we will use offtheshelf lowdensity parity check (LDPC) codes for pointtopoint communications as our basic code component. However, we emphasize the fact that this technique can be applied in conjunction with any linear codes, given appropriate modifications. Several works on the design of practical compute–forward strategies have also been considered in [OrdentlichZhanErezGastparNazer2011, HernNarayanan2013, FengSilvaKschischang2013, MejriOthman2015]. The main difference from the previous works is that we design a practical compute–forward strategy by explicitly using linear codes and the sumproduct decoding algorithm.
Our main contribution in this work is summarized as follows:

Following the theoretical insights developed in [LimFengPastoreNazerGastpar2016] for compute–and–forward using nested linear codes, we design nested LDPC codes based on offtheshelf LDPC codes for a practical implementation of CFMA.

Our decoding algorithm which is designed in a successive manner has complexity in the order of a conventional sumproduct decoder. In the first step of the decoding algorithm, we recover a linear combination of LDPC codewords using the sumproduct algorithm (SPA). In the next stage, we recover a codeword with another sumproduct decoder using the previously recovered linear combination as side information. In order to adapt the standard SPA in our scenario, we need to modify the initial input loglikelihood ratio (LLR) accordingly for each decoding step.

To support higher data rates using existing offtheshelf binary codes, we propose a multilevel binary LDPC code construction with higher order modulation. By appropriately modifying the input log likelihood ratio (LLR), we show that the standard SPA can be used in the decoding procedure. In particular, the standard SPA will be applied to every bit in the binary representation of the codeword separately with a modified initial input LLR, and the previously decoded bits will be used as side information for decoding the bits in the next level.

We further extend our technique to more complicated scenarios. Specifically, we discuss how to adapt our implementation to complexvalued channel models, to multipleaccess channels with more than two users, and to interference channels.
The remainder of the paper is organized as follows. In the next section we present our system model and discuss the theoretical background motivating our work. In section LABEL:sec:CFMA_basic the basic CFMA with binary codes is studied. In section LABEL:sec:CFMA_extensions extension of CFMA with multilevel binary codes is examined in details. In section LABEL:sec:CFMA_complex extension to complex channel is treated. Section LABEL:sec:simulation provides simulations results for various scenarios.
We use boldface lowercase and uppercase to denote vectors and matrices respectively. The operator ‘
’ denotes ary addition, and ‘’ (without subscript) is to be understood as binary addition (). The bracket denotes the set of integers .Ii System Model and Theoretical Background
We consider the twouser Gaussian MAC, with input alphabets , and output alphabet . For input symbols , , the output symbol is given by where denote the constant channel and is the additive white Gaussian noise.
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