A Design Criterion for the Rayleigh Fading Wiretap Channel Based on $\ell^1$-norm Theta Functions
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| Format: | Preprint |
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2024
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| _version_ | 1866916237789888512 |
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| author | Miller, Niklas |
| author_facet | Miller, Niklas |
| contents | 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 $Δ\approx 0.824858$ and kissing number 30, improving on the best known lattice packing density of the cross-polytope in dimension four. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04143 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Design Criterion for the Rayleigh Fading Wiretap Channel Based on $\ell^1$-norm Theta Functions Miller, Niklas Number Theory 11H06, 11F27, 11H31, 11H71 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 $Δ\approx 0.824858$ and kissing number 30, improving on the best known lattice packing density of the cross-polytope in dimension four. |
| title | A Design Criterion for the Rayleigh Fading Wiretap Channel Based on $\ell^1$-norm Theta Functions |
| topic | Number Theory 11H06, 11F27, 11H31, 11H71 |
| url | https://arxiv.org/abs/2405.04143 |