  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.
