Coding theorems via linear codes: Joint decoding rate regions

In several problems in network information theory, it is important to jointly decode the codewords transmitted by several encoders. For ensembles of i.i.d. random codes, such joint decoding can be analyzed directly with standard methods. In this paper, we develop a general framework to analyze joint decoding for ensembles of random linear codes and ensembles of nested linear codes. The technical difficulty stems from the fact that codewords are not generally independent. As an initial test case, we first present a joint decoding region for the standard multiple access problem using nested linear codes. Based on this result, we further establish a joint decoding rate region for compute-forward over the DM-MAC.