Physical-layer network coding (PNC) is a new approach to harnessing interference in wireless networks. Rather than avoiding interference or treating it as noise, PNC enables wireless relays to compute linear functions of the transmitted messages directly from the interfering signals. This allows for concurrent transmissions, thereby increasing network throughput.

This dissertation studies a family of the state-of-the-art PNC schemes called compute-and-forward (C). C was originally proposed and studied from an information-theoretic perspective. As such, it typically relies on several strong assumptions:​ very long block length, almost unbounded complexity, perfect channel state information, and no decoding errors at the relays;​ its benefit is often analyzed for simple network configurations. The aim of this dissertation is two-fold:​ first, to relax the above assumptions while preserving the performance of C, and second, to understand the benefit of C in more realistic network scenarios, such as random-access wireless networks.

There are four main results in this dissertation. First, an algebraicframework is developed, which establishes a direct connectionbetween C and module theory. This connection allows us tosystematically design lattice codes for C with controlled block length and complexity. In particular, explicit designcriteria are derived, concrete design examples are provided,and it is shown that nominal coding gains from 3 to 7.5dB can be obtained with relatively short block length and reasonable decodingcomplexity. Second, a new C scheme is proposed, which, unlike conventional C schemes, does not require any channel state information (CSI). It is shown that this CSI-free scheme achieves, for a certain class of lattice codes, almost the same throughput as its CSI-enabled counterpart. Third, an end-to-end error control mechanism is designed, which effectively mitigates decoding errors introduced at wireless relays. In particular, the end-to-end error control problem is modeled as a finite-ring matrix channel problem, for which tight capacity bounds and capacity-approaching schemes are provided. The final part of this dissertation studies the benefitof C in random-access wireless networks. In particular, it isshown that C significantly improves the network throughput and delay performance of slotted-ALOHA-based random-access protocols.