A Design Criterion for the Rayleigh Fading Wiretap Channel Based on $\ell^1$-norm Theta Functions

TL;DR

This paper establishes a new design criterion for Rayleigh fading wiretap channels using $\,\ell^1$-norm theta functions, providing bounds on decoding probabilities and constructing high-density lattice packings.

Contribution

It introduces a novel $\,\ell^1$-norm theta function framework for analyzing lattice codes in fading channels and develops algorithms for optimal lattice packings.

Findings

Bounds on eavesdropper decoding probability and legitimate error probability using theta functions.

Derived closed-form expressions for $\,\ell^1$-norm theta functions of sublattices.

Constructed a four-dimensional lattice with high packing density and improved kissing number.

Abstract

We show that the correct decoding probability of an eavesdropper and error probability of a legitimate receiver in a Rayleigh fading wiretap channel with lattice coset coding are both upper bounded by the theta function in the $\ell^1$-norm of the dual code lattices. Motivated by these findings, we derive a closed form expression for the $\ell^1$-norm theta function of any sublattice of $\mathbb{Q}^n$ and its dual, and prove that the lattice $\mathbb{Z}^n$ minimizes the theta function among all scalings of $\mathbb{Z}^n$ along the coordinate axes. Furthermore, we develop a method to algorithmically find locally critical lattice packings of the $n$-dimensional cross-polytope. Using this method, we are able to construct a four-dimensional lattice having a packing density of $\Delta\approx 0.824858$ and kissing number 30, improving on the best known lattice packing density of the cross-polytope in dimension four.