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

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.

20 pages