A Design Criterion for the Rayleigh Fading Wiretap Channel Based on \({^1}\)-Norm Theta Functions

.It is shown 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 bounded above by the theta function in the \(^1\)-norm of the dual code lattices. Motivated by this, we investigate the \(^1\)-norm theta functions of sublattices of \(Z^n\) and their duals and prove that the lattice \(Z^n\) minimizes the theta function among all scalings of \(Z^n\) along the coordinate axes. In the small \(q\)-range, the \(^1\)-norm theta function of a lattice is governed by the cross-polytope packing density. Thus, a method is developed 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 cross-polytope packing density of \(Δ ≈ 0.824858\) and kissing number 30, improving on the best known translational packing density of the cross-polytope in dimension four.Keywords\(^1\)-norm theta functionscross-polytope packingswiretap codeswireless communicationsMSC codes11H0611F2711H3111H71