A Joint Typicality Approach to Algebraic Network Information Theory

This paper presents a joint typicality framework for encoding and decoding nested linear codes for 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 the weighted sum of nested linear codewords 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, this framework can be used to shed light on the joint decoding rate region for compute–forward, which is considered an open problem. Specifically, this work establishes an achievable rate region for simultaneously decoding two linear combinations of nested linear codewords from K senders. Index Terms Linear codes, joint decoding, compute–forward, multiple-access channel, relay networks This paper was presented in part at the 2014 IEEE Information Theory Workshop, Hobart, Australia and the 2015 Allerton Conference on Communication, Control, and Computing, Monticello, IL. Sung Hoon Lim, Adriano Pastore, and Michael Gastpar are with the School of Computer and Communication Sciences, Ecole Polytechnique Fédérale, 1015 Lausanne, Switzerland (e-mail: [email protected], [email protected], [email protected]). Chen Feng is with the School of Engineering, The University of British Columbia, Kelowna, BC, Canada (e-mail: [email protected]). Bobak Nazer is with the Department of Electrical and Computer Engineering, Boston University, Boston, MA (e-mail: [email protected]).