Compute-Forward for DMCs: Simultaneous Decoding of Multiple Combinations

Abstract

Consider a receiver in a multiuser network that wishes to decode several messages. Simultaneous joint typicality decoding is one of the most powerful techniques for determining the fundamental limits at which reliable decoding is possible. This technique has historically been used in conjunction with random i.i.d. codebooks to establish achievable rate regions for networks. Recently, it has been shown that, in certain scenarios, nested linear codebooks in conjunction with "single-user" or sequential decoding can yield better achievable rates. For instance, the compute-forward problem examines the scenario of recovering L ≤ K linear combinations of transmitted codewords over a K-user multipleaccess channel (MAC), and it is well established that linear codebooks can yield higher rates. Here, we develop bounds for simultaneous joint typicality decoding used in conjunction with nested linear codebooks, and apply them to obtain a larger achievable region for compute-forward over a K-user discrete memoryless MAC. The key technical challenge is that competing codeword tuples that are linearly dependent on the true codeword tuple introduce statistical dependencies, which requires careful partitioning of the associated error events. Index Terms Compute-forward, joint decoding, linear codes, multiple-access channel I. INTRODUCTION For several decades, decode-forward [1], compress-forward [1], and amplify-forward [2] have served as the fundamental building blocks of transmission strategies for relay networks. These three relaying strategies were initially developed on canonical network models such as the relay channel and diamond relay network using random

A Discretization Approach to Compute–Forward

“…As a result, the limit arguments are considerably more subtle, and require a generalization of Rényi's d-dimensional entropy [8] to handle linear combinations (which we term algebraic entropy). Overall, this paper, alongside several recent works [6], [7], [9]- [14], demonstrates that algebraic approaches to network information theory problems can be handled via standard techniques, such as joint typicality encoding and decoding.…”

“…In this paper, we focus on the compute-forward problem where the goal is for one or more receivers to recover linear combinations of the transmitters' messages. Prior work has derived achievable rate regions for the special case of Gaussian channels using nested lattice codes [4], [5] and for the special case of discrete memoryless channels using nested linear codes [6], [7]. In particular, the latter approach employs standard joint typicality encoding and decoding techniques.…”

“…Specifically, our approach recovers prior Gaussian compute-forward results [4], [5], and, in some cases, improves upon them. This is due to the fact that the underlying rate region for discrete channels [7] is based on simultaneous decoding rather than sequential decoding.…”

Vandermonde-Based Unitary Precoding Method for Integer-Forcing Design

“…Compute-and-forward (CF) is a promising new technique of physical-layer network coding for wireless relay networks [1][2][3]. CF exploits nested lattices and enables relays to decode the integer linear combination of transmitted messages by using the noisy linear combinations provided by the channel.…”

Intelligent Reflecting Surfaces for Compute-and-Forward

“…The proposed scheme in [15], based on nested lattice codes, provides a framework to do this, and furthermore determines the maximum transmission rate of the users so that such computation is feasible, namely, the computation rate. After publication of [15] many research works have benefited from this framework in different communication theory problems (e.g., see [16], [17] and [18]).…”