Lattice Index Coding

TL;DR

This paper introduces lattice index codes for noisy Gaussian broadcast channels, leveraging lattice structures and the Chinese remainder theorem to optimize side information utilization and improve communication efficiency.

Contribution

It proposes a novel lattice index coding scheme using various algebraic lattices and introduces the side information gain metric to evaluate code performance.

Findings

Codes achieve maximum side information gain among densest lattices.

Construction methods include rational, Gaussian, Eisenstein, and Hurwitz quaternion lattices.

Example demonstrates benefits in Gaussian broadcast channels with complex demands.

Abstract

The index coding problem involves a sender with K messages to be transmitted across a broadcast channel, and a set of receivers each of which demands a subset of the K messages while having prior knowledge of a different subset as side information. We consider the specific case of noisy index coding where the broadcast channel is Gaussian and every receiver demands all the messages from the source. Instances of this communication problem arise in wireless relay networks, sensor networks, and retransmissions in broadcast channels. We construct 'lattice index codes' for this channel by encoding the K messages individually using K modulo lattice constellations and transmitting their sum modulo a coarse lattice. We introduce a design metric called 'side information gain' that measures the advantage of a code in utilizing the side information at the receivers, and hence its goodness as an index code. Based on the Chinese remainder theorem, we then construct lattice index codes with large side information gains using lattices over the following principal ideal domains: rational integers, Gaussian integers, Eisenstein integers, and the Hurwitz quaternions. Among all lattice index codes constructed using any densest lattice of a given dimension, our codes achieve the maximum side information gain. Finally, using an example, we illustrate how the proposed lattice index codes can benefit Gaussian broadcast channels with more general message demands.