Probability Estimates for Fading and Wiretap Channels from Ideal Class Zeta Functions

TL;DR

This paper develops new probability estimates for ideal lattice codes from totally real number fields, incorporating ideal class Dedekind zeta functions, and extends previous work by removing the principal ideal assumption.

Contribution

It introduces a novel approach to estimate inverse norm sums for ideal lattices using ideal class Dedekind zeta functions, accounting for non-principal ideals.

Findings

Inverse norm sum depends on regulator, discriminant, and Dedekind zeta values.

Derived estimates for the number of elements with a given algebraic norm.

Examples demonstrate the accuracy and predictive power of the theorems.

Abstract

In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, but also on the values of the ideal class Dedekind zeta functions. Along the way, we derive an estimate of the number of elements in a given ideal with a certain algebraic norm within a finite hypercube. We provide several examples which measure the accuracy and predictive ability of our theorems.