A Joint Typicality Approach to Compute–Forward

This paper presents a joint typicality framework for encoding and decoding nested linear codes in multi-user networks. This framework provides a new perspective on compute–forward within the context of discrete memoryless networks. In particular, it establishes an achievable rate region for computing a linear combination over a discrete memoryless multiple-access channel (MAC). When specialized to the Gaussian MAC, this rate region recovers and improves upon the lattice-based compute–forward rate region of Nazer and Gastpar, thus providing a unified approach for discrete memoryless and Gaussian networks. Furthermore, our framework provides some valuable insights on establishing the optimal decoding rate region for compute–forward by considering joint decoders, progressing beyond most previous works that consider successive cancellation decoding. Specifically, this paper establishes an achievable rate region for simultaneously decoding two linear combinations of nested linear codewords from $K$ senders.

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