Design of Network Codes for Multiple-User Multiple-Relay Wireless Networks

We investigate the design of network codes for multiple-user multiple-relay (MUMR) wireless networks with slow fading (quasi-static) channels. In these networks, M users have independent information to be transmitted to a common base station (BS) with the help of N relays, where M >= 2 and N >= 1 are arbitrary integers. We investigate such networks in terms of diversity order to measure asymptotic performance. For networks with orthogonal channels, we show that network codes based on maximum distance separable (MDS) codes can achieve the maximum diversity order of N + 1. We further show that the MDS coding construction of network codes is also necessary to obtain full diversity for linear finite field network coding (FFNC). Then, we compare the performance of the FFNC approach with superposition coding (SC) at the relays. The results show that the FFNC based on MDS codes has better performance than SC in both the high rate and the high SNR regime. Further, we discuss networks without direct source-to-BS channels for N >= M. We show that the proposed FFNC can obtain the diversity order N - M + 1, which is equivalent to achieving the Singleton bound for network error-correction codes. Finally, we study the network with nonorthogonal channels and show our codes can still achieve a diversity order of N + 1, which cannot be achieved by a scheme based on SC.