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

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 ¹-norm of the dual code lattices. Motivated by these findings, we derive a closed form expression for the ¹-norm theta function of any sublattice of Qⁿ and its dual, and prove that the lattice Zⁿ minimizes the theta function among all scalings of Zⁿ 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 Δ≈ 0.824858 and kissing number 30, improving on the best known lattice packing density of the cross-polytope in dimension four.