Abstract
In this chapter we will review classical and recent advances on “probabilistic” constructions for Euclidean lattices. We will then show recent refinements of these techniques using algebraic number theory. The interest in algebraic lattices is twofold: on the one hand, they are key elements for the construction of sphere packings with the best known asymptotic density; on the other hand, they provide effective solutions to a number of wireless communication problems. We will focus on applications to fading channels, multiple-input-multiple-output (MIMO) channels and to information-theoretic security.
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Notes
- 1.
The square of this number is known as the Hermite constant.
- 2.
Notice that in this part we consider complex lattices, since \(\mathbb {C}^T\) is the typical ambient space in applications to wireless. The results in the previous sections can be “adapted” to complex lattices in a natural way. For instance, a complex full-rank lattice in \(\mathbb {C}^T\) can be naturally identified with a lattice in \(\mathbb {R}^{n}\), for n = 2T.
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Campello, A., Ling, C. (2020). Random Algebraic Lattices and Codes for Wireless Communications. In: Beresnevich, V., Burr, A., Nazer, B., Velani, S. (eds) Number Theory Meets Wireless Communications. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-61303-7_4
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