Lattice Index Codes from Algebraic Number Fields

TL;DR

This paper extends lattice index coding strategies from principal ideal domains to general algebraic number fields, enhancing design flexibility and achieving side information and diversity gains in broadcast channels.

Contribution

It generalizes lattice index coding to algebraic number fields beyond PIDs, providing bounds on side information gains and demonstrating diversity benefits over Rayleigh fading channels.

Findings

Bounds on side information gains are asymptotically tight.

The scheme includes the original PID-based approach as a special case.

Number field-based codes offer diversity gains in fading channels.

Abstract

Broadcasting $K$ independent messages to multiple users where each user demands all the messages and has a subset of the messages as side information is studied. Recently, Natarajan, Hong, and Viterbo proposed a novel broadcasting strategy called lattice index coding which uses lattices constructed over some principal ideal domains (PIDs) for transmission and showed that this scheme provides uniform side information gains. In this paper, we generalize this strategy to general rings of algebraic integers of number fields which may not be PIDs. Upper and lower bounds on the side information gains for the proposed scheme constructed over some interesting classes of number fields are provided and are shown to coincide asymptotically in message rates. This generalization substantially enlarges the design space and partially includes the scheme by Natarajan, Hong, and Viterbo as a special case. Perhaps more importantly, in addition to side information gains, the proposed lattice index codes benefit from diversity gains inherent in constellations carved from number fields when used over Rayleigh fading channel. Some interesting examples are also provided for which the proposed scheme allows all the messages to be from the same field.