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arXiv · 2405.04143

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

Niklas Miller
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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.

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Niklas Miller. 2024-05-07. A Design Criterion for the Rayleigh Fading Wiretap Channel Based on $\ell^1$-norm Theta Functions. https://arxiv.org/abs/2405.04143

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