Towards an algebraic network information theory: Simultaneous joint typicality decoding

Recent work has employed joint typicality encoding and decoding of nested linear code ensembles to generalize the compute-forward strategy to discrete memoryless multiple-access channels (MACs). An appealing feature of these nested linear code ensembles is that the coding strategies and error probability bounds are conceptually similar to classical techniques for random i.i.d. code ensembles. In this paper, we consider the problem of recovering K linearly independent combinations over a K-user MAC, i.e., recovering the messages in their entirety via nested linear codes. While the MAC rate region is well-understood for random i.i.d. code ensembles, new techniques are needed to handle the statistical dependencies between competing codeword K-tuples that occur in nested linear code ensembles.