In this paper, we propose a network coding scheme for up-link wireless backhaul networks to increase the throughput while guaranteeing full diversity order. The key advance proposed is the employment of precoding complex field matrices for two user and two base station backhaul networks to achieve throughput of 1 symbol/channel use while guaranteeing full diversity order for this one-way two-hop communications setting.

In particular, the throughput of the considered wireless backhaul network using the proposed network coding scheme is 50% higher than that using conventional complex field network coding (CFNC) and 200% higher than that using conventional amplify-and-forward (AF) relay transmission. Importantly, the considered network obtains full diversity order, i.e., the diversity order is the same as that of the respective conventional CFNC and AF relay networks. Numerical results indicate that the simulation results of the pairwise error probability tightly converge to the asymptotic curves in the high signal-to-noise ratio.