Compute-and-Forward (CoF) is an innovative physical layer network coding strategy, designed to enable receivers in wireless communications to effectively utilize interference. The key idea of CoF is to implement integer combinations based on the codewords from multiple transmitters, rather than decoding individual source codewords. Although CoF is widely used in wireless relay networks, there are still some problems to be solved, such as rank failure, single antenna reception, and the shortest vector problem. In this paper, we introduce a successive extended CoF (SECoF) as a pioneering solution tailored for multi-source, multi-relay, and multi-antenna wireless relay networks. First, we analyze the traditional CoF, and design a SECoF method combining the concepts of matrix projection and successive interference cancellation, which overcomes the problem of CoF rate tending to zero and rank failure and improves the network performance. Secondly, we obtain an approximate solution to the integer-value coefficient vectors by using the LLL lattice-based resolution algorithm. In addition, we deduce the corresponding concise formulas of SECoF. Simulation results show that the SECoF has strong robustness and the approaches outperform the state-of-the-art methods in terms of computation rate, rank failure probability, and outage probability.
Keywords: compute-and-forward; matrix projection algorithm; rank failure; successive extended computation; wireless relay network.