Wireless Network-Coded Multi-Way Relaying Using Latin Hyper-Cubes

Physical layer network-coding for the n-way wireless relaying scenario is dealt with, where each of the n user nodes X1, X2, ..., Xn wishes to communicate its messages to all the other (n−1) nodes with the help of the relay node R. The given scheme, based on the denoise-and-forward scheme proposed for two-way relaying by Popovski et al. in [1], employs two phases: Multiple Access (MA) phase and Broadcast (BC) phase with each phase utilizing one channel use and hence totally two channel uses. Physical layer network-coding using the denoise-and-forward scheme was done for the two-way relaying scenario in [2], for three-way relaying scenario in [3], and for four-way relaying scenario in [11]. This paper employs denoise-and-forward scheme for physical layer network coding of the n-way relaying scenario illustrating with the help of the case n = 5 not dealt with so far. It is observed that adaptively changing the network coding map used at the relay according to the channel conditions reduces the impact of multiple access interference which occurs at the relay during the MA phase. These network coding maps are chosen so that they satisfy a requirement called exclusive law. We show that when the n users transmit points from the same M -PSK (M = 2) constellation, every such network coding map that satisfies the exclusive law can be represented by a n-fold Latin Hyper-Cube of side M . The singular fade subspaces resulting from the scheme are described and enumerated for general values of n and M and are classified based on their removability in the given scenario. A network code map to be used by the relay for the BC phase aiming at reducing the effect of interference at the MA stage is obtained. I. BACKGROUND AND PRELIMINARIES The two-stage protocol for physical layer network coding for the two-way relay channel first introduced in [4], exploits the multiple access interference occurring at the relay so that the communication between the end nodes can be done using a two stage protocol. The works in [5], [6] deal with the information theoretic studies for bidirectional relaying. In [2], modulation schemes to be used at the nodes for uncoded transmission for the two-way relaying were studied. The work done for the relay channels with three or more user nodes is given in [3], [7]–[11]. In [7], authors have proposed a two stage operation for three-way relaying called joint network and superposition coding, in which the three users transmit to the relay node one-by-one in the first phase, and the relay node makes two superimposed XOR-ed packets and transmits back to the users in the BC phase. The packet from the node with the worst channel gain is XOR-ed with the other two packets. The protocol employs four channel uses, three for the MA phase and one for the BC phase. It is claimed by the authors that this scheme can be extended to more Fig. 1. An n-way relay channel than three users as well. In the work by Pischella and Ruyet in [8] a lattice-based coding scheme combined with power control, composed of alternate MA and BC phases, consisting of four channel uses for three-way relaying is proposed. The relay receives an integer linear combination of the symbols transmitted by the user nodes. It is stated that the scheme can be extended to more number of users. These two works essentially deal with the information theoretic aspects of multiway relaying. An ‘opportunistic scheduling technique’ for physical network coding is proposed by authors Jeon et al. in [10], where using a channel norm criterion and a minimum distance criterion, users in the MA as well as the BC phase are selected on the basis of instantaneous SNR. This approach utilizes six channel uses in case of three-way relaying and it is mentioned that the approach can be extended to more number of users. In [9], a ‘Latin square-like condition’ for the three-way relay channel network code is proposed and cell swapping techniques on these Latin Cubes are suggested in order to improve upon these network codes. The protocol employs five channel uses, and the channel gains associated with the channels are not considered in the construction of this network coding map. We consider the n-way wireless relaying scenario shown in Fig. 1, where n-way data transfer takes place among the nodes X1, X2,..., Xn with the help of the relay R assuming that the n nodes operate in half-duplex mode. The relaying protocol consists of two phases, multiple access (MA) phase, consisting of one channel use during which X1, X2,..., Xn transmit to R; and broadcast (BC) phase, in which R transmits to X1, X2,..., Xn in a single channel use. Network Coding is employed at R in such a way that Xi can decode Xj’s message for i, j = 1, 2, ..., n and j 6= i, given that Xi knows its own message. Latin Cubes have been explored as a tool to find the network coding map used by the relay, depending on the channel gain ar X iv :1 30 3. 02 29 v1 [ cs .I T ] 1 M ar 2 01 3 in [3]. The throughput performance of the two stage protocol for three-way relaying given in [3] is better than the throughput performance of the ‘opportunistic scheduling technique’ given in [10] at high SNR, as can be observed from the plots given in [3]. The work in [11] further extends the approach used in [3] to four-way relaying and employs two channel uses for the entire information exchange amongst the four users, which makes the throughput performance of the scheme better than the other existing schemes. This scheme that utilizes two channel uses for the entire information exchange between three and four users using a relay in [3] and [11] respectively, is extended to n users in this paper, for the case when M -PSK is used at the end nodes. For our physical layer network coding strategy we use the mathematical structure called a Latin Hyper-Cube defined as follows: Definition 1: An n-fold Latin Hyper-Cube L of r-th order of side M [12] is an M × M × ... × M (n times) array containing M entries, Mn−r of each of M kinds, such that each symbol occurs at most once for each value taken by each dimension of the hyper-cube. 1 For our purposes, we use only n-fold Latin Hyper-Cubes of side M on the symbols from the set Zt = {0, 1, 2, ..., t− 1}, t ≥Mn−1.