On code design in joint MAC scheduling and wireless network coding

In this paper, we investigate the problem of network coding and media scheduling in wireless multihop networks. Unique characteristics of the wireless media, such as omnidirectional transmissions and destructive interference, as well as the limitations of wireless nodes to either transmit or receive at any given time, imply new code design constraints for wireless networks. In this work, following the approach of Sagduyu et al., given a sufficient set of conflict-free link sets, we investigate necessary and sufficient conditions to design a capacity-achieving network coding solution for wireless networks. This approach uses a wired graph representation of the wireless network to design wireless network codes. Our study shows that the proposed approach works only if we preserve all of the code characteristics of the wired graph in the wireless one. We show the shortcoming of previous studies through a counterexample and explore this case further. The resulting insight is summarized in Theorem 1, which proves that unequal time shares allocated to conflict-free link sets will result in changing coding coefficients as the sets with larger time share are scheduled repeatedly.