Nonasymptotic Probability Bounds for Fading Channels Exploiting Dedekind Zeta Functions

TL;DR

This paper introduces new nonasymptotic probability bounds for algebraic lattice codes in fading channels by leveraging Dedekind zeta functions, providing practical estimates for error performance and security in finite-dimensional settings.

Contribution

It applies Dedekind zeta functions to derive novel probability bounds for lattice codes in fading channels, bridging algebraic number theory and communication theory.

Findings

Upper bounds on error probability for finite constellations

Estimates of algebraic elements within hyper-cubes

Application to wiretap channel security

Abstract

In this paper, new probability bounds are derived for algebraic lattice codes. This is done by using the Dedekind zeta functions of the algebraic number fields involved in the lattice constructions. In particular, it is shown how to upper bound the error performance of a finite constellation on a Rayleigh fading channel and the probability of an eavesdropper's correct decision in a wiretap channel. As a byproduct, an estimate of the number of elements with a certain algebraic norm within a finite hyper-cube is derived. While this type of estimates have been, to some extent, considered in algebraic number theory before, they are now brought into novel practice in the context of fading channel communications. Hence, the interest here is in small-dimensional lattices and finite constellations rather than in the asymptotic behavior.