Abstract-The design of modulation schemes for the physical layer network-coded two-way relaying scenario is considered with a protocol which employs two phases: multiple access (MA) phase and broadcast (BC) phase. It was observed by Koike-Akino et al. that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of MA interference which occurs at the relay during the MA phase and all these network coding maps should satisfy a requirement called the exclusive law. We show that every network coding map that satisfies the exclusive law is representable by a Latin Square and conversely, that this relationship can be used to get the network coding maps satisfying the exclusive law. The channel fade states for which the minimum distance of the effective constellation at the relay become zero are referred to as the singular fade states. For PSK modulation ( any power of 2), it is shown that there are singular fade states. Also, it is shown that the constraints which the network coding maps should satisfy so that the harmful effects of the singular fade states are removed, can be viewed equivalently as partially filled Latin Squares (PFLS). The problem of finding all the required maps is reduced to finding a small set of maps for PSK constellations ( any power of  E consider the two-way wireless relaying scenario shown in Fig. 1 , where bidirectional data transfer takes place between the nodes A and B with the help of the relay R. It is assumed that all three nodes operate in half-duplex mode, i.e., they cannot transmit and receive simultaneously in the same frequency band. The relaying protocol consists of the following two phases: the multiple access (MA) phase, during which A and B simultaneously transmit to R and the broadcast (BC) phase during which R transmits to A and B. Network coding is employed at R in such a way that A (B) can decode the message of B (A), given that A (B) knows its own message. The concept of physical layer network coding has attracted a lot of attention in recent times. The idea of physical layer network coding for the two way relay channel was first introduced in [3] , where the MA interference occurring at the relay was exploited so that the communication between the end nodes can be done using a two phase protocol. Information theoretic studies for the physical layer network coding scenario were reported in [4] and [5]. A differential modulation scheme with analog network coding for bidirectional relaying was proposed in [6] . The design principles governing the choice of modulation schemes to be used at the nodes for uncoded transmission were studied in [7] . An extension for the case when the nodes use convolutional codes was done in [8] . A multilevel coding scheme for the two-way relaying scenario was proposed in [9] . Power allocation strategies and lattice-based coding schemes for bidirectional relaying were proposed in [10] . It was observed in [7] that for uncoded transmission, the network coding map used at the relay needs to be changed adaptively according to the channel fading coefficients, in order to minimize the impact of the MA interference. In this paper, it is shown that a finite number of network coding maps, obtained by completion of partially filled Latin Squares (PFLS), can effectively mitigate the effect of MA interference. Having obtained the network coding maps, the set of all possible channel realizations is quantized into a finite number of regions, with a specific network coding map chosen at R in a particular region. In [7] , a quantization was obtained using exhaustive computer search for 4-PSK signal set. In this paper, we obtain a quantization analytically for any -PSK signal set, for any power of 2.
Nonlinear PNC mapping based on Latin square was proposed in REF .
Wireless Network-Coded Bidirectional Relaying Using Latin Squares for $M$ -PSK Modulation
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Computer Science