# Analysis and Design of Rateless Two-way Relay Networks Based on a Multiply-and-Forward Scheme
## 1 Introduction  
Over the past few decades, the two-way relay network (TWRN) [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13] has aroused great interest due to the overwhelming development of wireless cellular systems, especially when one-way relaying is of low spectral efficiency because of the half-duplex constraint. In a TWRN system, two source nodes exchange information through a relay node. A TWRN can be realized in three or even two stages with the help of network coding technologies. In the three-stage scheme, source nodes send their corresponding messages to the relay in the first two stages, respectively. In the third stage, the relay processes received signals with either a decode-and-forward (DF) approach [1; 2; 3; 4; 5; 6] or an amplify-and-forward approach [7; 8; 9; 10; 11; 12; 13], and then broadcasts back to the source nodes. In [14], [PERSON] et al. proposed a multiply-and-forward (MF) approach, in which the relay multiplies the signals sent from different source nodes. More importantly, MF has shown a significant performance increase compared to AF in a three-stage TWRN system [14; 15; 16; 17].  
To ensure reliable data transmission in a large-scale network, fountain codes [18; 19; 20] were proposed and have been well investigated in binary erasure channels and noisy channels [21; 22]. Some works have been carried out on designing well-performing degree distributions in noisy channels [22; 23; 24]. [PERSON] et al. proposed the concept of systematic Luby transform (SLT) codes and put forth the truncated robust soliton degree distribution for SLT codes in wireless channels [23]. [PERSON] et al. [24] extended the work of [PERSON] et al. [22] for raptor codes in memoryless binary symmetric channels to study systematic LT codes in binary input additive white Gaussian noise (BIAWGN) channels and analyzed the asymptotic bit error rate (BER) performance using a Gaussian approximation technique.  
To take advantage of relaying systems and rateless codes, we therefore consider employing fountain codes, a class of rateless codes in the TWRN for channel coding at source nodes in this paper. [PERSON] et al. [25] analyzed a three-stage rateless coded protocol for a DF-TWRN and proposed somerelated optimization models for rateless codes used at source nodes. Since the relay carries out decoding operations in the DF approach, which is more complex than AF and MF, we focus on fountain-coded two-way MF relaying in this paper. Specifically, we consider the SLT-coded MF-TWRN over Rayleigh flat fading channels. Furthermore, we will investigate BER performance of our proposed MF-TWRN and its corresponding lower bound on BER. In addition, an optimization model will be derived to generate the well performing degree distribution for SLT codes in the MF-TWRN. Our proposed rateless relaying scheme can lead to heuristic designs of TWRN extending the range of wireless transmission, such as in beyond-5G cellular systems and wireless ad hoc networks.  
The rest of this paper is organized as follows. Section 2 introduces the system model of three-stage TWRN and Section 3 derives and compares performance of uncoded TWRN with MF and AF schemes. In Section 4, we establish the equivalent point-to-point channel model, and further study BER performance as well as the optimization method for SLT-coded MF-TWRN. Some numerical results are shown in Section 5, and Section 6 concludes the paper.  
The key variables and notations used in this paper are summarized in Table 1.